Method for extracting spatial knowledge from an input signal using computational manifolds

ABSTRACT

A processor architecture for a learning machine is presented which uses a massive array of processing elements having local, recurrent connections to form global associations between functions defined on manifolds. Associations between these functions provide the basis for learning cause-and-effect relationships involving vision, audition, tactile sensation and kinetic motion. Two arbitrary input signals hold each other in place in a manifold association processor and form the basis of short-term memory.

RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/749,076, filed May 15, 2007, which claims the benefit of the priorityof U.S. Provisional Applications No. 60/801,026, filed May 16, 2006, andNo. 60/862,038, filed Oct. 18, 2006, each of which is incorporatedherein by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates to a physiologically realistic model forneurons and neurotransmitters and use of such a model in image andsignal analysis and for data storage.

BACKGROUND OF THE INVENTION

Since its inception over fifty years ago, the field of artificialintelligence has remained separated from the field of psychology by thefundamental mathematical dichotomy between the discrete and thecontinuous. While computer scientists regard symbols as discrete bitpatterns, psychologists regard them as continuous visual icons. Thesubstitution of integral signs for summation signs sometimes serves tocreate a bridge across this dichotomy, for example, between thecontinuous and discrete versions of the Fourier transform.

It is easier to design a symbol processing system on a digital computerby representing symbols as discrete bit patterns, than to design asystem that represents symbols as cumbersome digitized images. Inbiological systems, the central nervous system (CNS) represents andoperates on symbols as continuous images because it was forced to do so.Throughout the course of evolution, everything in nature, predator andprey, the external environment and the body itself, was governed by thelaws of physics, which are based on functions defined in space and time.In the struggle to survive, an organism had to process, as quickly aspossible, continuous symbol representations.

The lack of a complete and thorough scientific understanding of animalintelligence and human consciousness has limited the ability to buildcomputer systems that mimic these capabilities. The present inventionprovides a new model that may provide insight into how to design suchsystems. The design connects together a massive number of relativelysmall processing elements to form a large recurrent image associationsystem. It is completely parallelizable, and all of the computationstake place recursively over small, localized regions.

The mathematical definition of a manifold is a metric space, which iseverywhere locally homeomorphic to an open set in R^(n), where R denotesthe real numbers (Spivak, 1979; Schutz, 1980). Less formally, we canregard a one-dimensional manifold as a smooth curve, a two-dimensionalmanifold as a smooth surface, akin to a deformed balloon, andanalogously for higher order manifolds. In order to accommodatediscontinuities in the real world, we use Borel measurable functions torepresent data, and define operations using Lebesgue integrals(Kolmogorov & Fomin, 1970; Royden, 1988). For brevity, in the followingdiscussion, we will refer to generalized or Borel measurable functionssimply as functions.

Since open regions of space, time and frequency as well as their productspaces are all manifolds, one can accurately describe virtuallyeverything in nature, as a function defined on a manifold. At the levelof quantum physics, matter and energy are discrete. Nevertheless, as isthe case for the electrons that define the current density J inMaxwell's equations, and the molecules that define the density of matterρ in the Navier-Stokes equations, at the macroscopic level thesequantities are differentiable.

BRIEF SUMMARY OF THE INVENTION

The inventive model differs from previous work on neural fields in thatit does not attempt to describe the mean firing rate of neurons as acontinuous field. Instead, a model is described in which the dendritictree of a neuron behaves as if the concentration of neurotransmitterssurrounding it in the extracellular space is a continuous field.

According to the present invention a neural model is constructed fromtwo stateless logic gates, and incorporates a recursive feedback loopwhere a single bit and its complement hold each other in place. Afundamental unit of digital memory, the SR flip-flop, is capable ofstoring a single bit of information. The invention defines a continuousversion of this unit of memory which uses two reciprocal images to holdeach other in place using an analogous recursive feedback loop. TheBrodmann areas, the constituent parts of the cerebral cortex, arephysiological realizations of these reciprocal image memories.

For brevity, we use the term image to denote a scalar or vector valuedfunction defined on a two-dimensional manifold, for example an open,connected region of a plane. In the realm of natural computation, thisgeneral notion of an image plays a role that extends beyond computervision. For example, an audio spectrogram that represents time in thehorizontal dimension and frequency in the vertical dimension is atwo-dimensional image. Similar tonotopic maps exist at several locationsin the CNS. In this context, learning and retrieving image associationsis a basic cognitive function. For example when a child learns a newword, say “horse”, they learn an association between two images: thevisual image of a horse and the two-dimensional audio spectrogram of thesound of the word “horse”.

The generalization of images to include any function defined on atwo-dimensional manifold is a powerful one in terms of its ability todescribe and model many known neuroscientific phenomena. For example,the surface of the skin is a two-dimensional somatosensory image and itslocation in three-dimensional space is a vector valued function definedon the same image domain. The union of the cross-sections of all musclesforms a somatotopic image.

We can easily extend all of the algorithms presented here from images togeneralized functions defined on any n-dimensional manifold. Therefore,we use the term Manifold Association Processor (MAP) to denote a devicethat is capable of directly forming and retrieving associations betweenimages.

The exemplary embodiments include three types of Manifold AssociationProcessors. The first two MAP descriptions are abstract and areintentionally defined only in terms of their external, observablebehavior. The first of these, the Λ-MAP model, is stateless but canperform logic operations by producing the output image that isassociated with multiple input images. Two recursively-connected Λ-MAPsform the SR-MAP, which is analogous to a set-reset (SR) flip-flop wherewe replace the individual bits with two-dimensional images and the NANDgates with Λ-MAPs. The Ψ-MAP description is concrete and specifies theinternal principles of operation. Its unique design uses recurrentinterconnections to integrate the results of many locally connectedprocessing elements and thereby form an overall global associationbetween two arbitrary images.

Borrowing from the terminology of digital design, specifically aprogrammable logic array, we use the term Ψ-MAP Array to designate aninterconnected collection of Ψ-MAPs where the bits are again replaced byimages. A Ψ-MAP is a model of a single Brodmann area. An array ofinterconnected Ψ-MAPs models the entire collection of Brodmann areasthat make up the neocortex.

Without recurrence, locally connected neurons could not establishintelligible and meaningful associations on a global scale. However, thespectral representation of data allows a massive number of parallelprocessors, with local recursive connections, to operate in concert andfind the associations between the reciprocal images that constitute thecore of the Brodmann areas. The union of these mutually reinforcingimage pairs covers the entire cerebral cortex and is the basis ofshort-term memory. This memory, combined with the ability to focusattention on select regions and search for new associations provides thefoundation that underlies cognition and rational thought.

Random coincidences do happen and we must be careful not to impart toomuch importance to a chance occurrence. However, so many correlationsbetween a new model and previously unexplained, general neurologicalfacts, suggests the need for additional analysis and furtherconsideration.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be more clearly understood from the followingdetailed description of the preferred embodiments of the invention andfrom the attached drawings, in which:

FIG. 1 illustrates the abstract Logic Manifold Association Processorproducing an output image from input images.

FIGS. 2 a and 2 b illustrates a logic circuitry of a fundamental unit ofmemory where FIG. 2 a illustrates two inverters and FIG. 2 b illustratestwo NAND gates forming a SR flip-flop.

FIG. 3 is a flow diagram of a Set-Reset Manifold Association Processor.

FIG. 4 illustrates processing elements taking pixels from input imagesproducing an output image.

FIG. 5 is a diagrammatic view of a feedback loop in the Set-ResetManifold Association Processor.

FIG. 6 is a schematic view of the Set-Reset Manifold AssociationProcessor Array model.

FIG. 7 provides an isomorphic mapping of functions.

FIG. 8 illustrates the realization of a neural manifold at the cellularlevel.

FIG. 9 a illustrates a prior art neural network; FIG. 9 b is a cellularlevel illustration of manifolds, the hidden layer, and processingelements; and FIG. 9 c illustrates a continuous neural manifold model.

FIG. 10 is a diagrammatic view of the detailed Set-Reset ManifoldAssociation Processor computational model illustrating a general patternof interconnection.

FIG. 11 is a diagrammatic view of the Set-Reset Manifold AssociationProcessor learning pathways.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Definitions: For purposes of the description of the invention, thefollowing definitions are used:

A “neighborhood” of x is an open set containing x.

A “manifold” is a metric space M with the property that for every x in Mthere is a neighborhood U of x and an integer n≧0 such that U ishomeomorphic to R^(n). Examples of manifolds include open subsets oftime, space and frequency as well as their product spaces. Examples oftwo-dimensional manifolds (n=2) include the plane, a disk, the surfaceof a sphere or the surface of a torus.

The term “measure” is assumed to mean a “signed measure” that is onethat can take negative values. A signed measure can be evaluated as thedifference of two nonnegative measures.

The terms “image” and “signal” refer to any scalar or vector-valuedfunction defined on any manifold of any dimension. For purposes of theinstant description, the terms “image” and “signal” may be usedinterchangeably, such that a description of steps for processing animage may be applied equally to processing of a signal, and vice versa.

A “neural manifold” is the continuous counterpart to a neural network.It defines a mapping between functions defined on the input manifold tofunctions defined on the output manifold. The mathematics of neuralnetworks generalizes to neural manifolds the way the inner product in aHilbert space applies equally well to either the sum of the products ofthe coefficients of two vectors (neural network) or to the integral ofthe product of two functions (neural manifold). We extend the termneural manifold from the case of a finite number of weights to the caseof an infinite number of weights corresponding to the values of afunction defined on a manifold.

A “pixel” or “voxel” signifies a discretized sample taken from an imageor signal. The sampling of the manifold may be on a grid which isregular or irregular, that is the pixels need not represent areas whichare uniform in size or shape. A pixel may be a vector of real values,e.g. red, green and blue channels of a color photograph. The componentsof the vector may also represent averages or derivatives over regions ofthe images.

A “processing element” is a computational unit similar to a neuron. Itis not intended to mean a digital computing element such as amicroprocessor. A single microprocessor or CPU (central processing unit)may simulate the computations of many processing elements.

A “null image” is a predefined image that is regarded as blank orcontaining no information. For example, an image that was completelywhite, black or some predefined shade of grey could be used as a nullimage.

“Image representation” is a general term used to signify either an imageor its isomorphic representation. The isomorphic representation may, forexample, be the result of applying a spectral transform, such as theFourier transform or the wavelet transform, to the image.

The “solution space” is a subset of the set of all possible images, orimage spectra that correspond to correct or valid images.

“Reciprocal images” or “image pair” means two images defined on twomanifolds where each image when applied as an input to one of the twoΛ-MAPs produces the other as an output. The two manifolds may or may nothave similar types and dimensions. The two images are referred as thefirst and second images of an image pair.

A “function space” is a set of functions that all have the same domainand range (co-domain). Specific functions may be referred to as “points”or “elements” in a function space.

“Encoding” is defined as a transformation of functions. Mathematically,it is a mapping from one function space to another function space.

“Topological alignment” refers to a combination of many functions thatall have the same domain (input manifold). The outputs of the multiplefunctions are combined by constructing a new vector-valued function thatmaps inputs to the Cartesian product space of the individual functionco-domains.

A collection of reciprocal image pairs is a set of reciprocal imageswhere the functions that make up the first image of each pair all havethe same domain and range, and the functions that make up the secondimages also all have the same domain and range.

To facilitate understanding of the present invention, it is helpful todescribe some of the relevant research that is taking place, and compareit with the Ψ-MAP algorithms of the invention by looking at some oftheir important characteristics and analyzing them sequentially. Thefirst of these characteristics is the process of converting logic intomemory.

Willshaw (1969) first proposed a model of associative memories usingnon-holographic methods. By definition, any function, which mapselements of one set to elements of another set, performs an associationoperation and is therefore an abstract associative memory. The abilityof neural networks to approximate functions allows them to be used asconcrete associative memory implementations. Recurrent neural networks(RNNs) overcome some of the limitations of feed-forward designs andmodel the characteristics of a system over time.

A second type of associative memory results from following thetrajectory of a particle in a dynamical system. A simple recurrence ofthe form x_(i+1)=f (x_(i)) is the basis for RNN associative memorieswhere the association is between an initial state and the final statedetermined by recursive iterations. Consequently, the association is nota fixed static relationship, but a trajectory representing the timebehavior of a state variable as it moves through a multidimensionalspace. To distinguish between these two types of association mechanismswe will refer to the second type, which follows the trajectory of apoint in a dynamical system, as a classification. Each unique fixedpoint identifies a class that equals the set of all points whosetrajectory leads to that point.

Direct application of this technique has been used to create imageclassification systems. Dynamic-system associations formed in thespectral domain can reduce noise in communication applications.

The network weights, learned during the training process, act aslong-term memory capable of recalling associations, but cannot serve asthe working memory required for computations. Short-term or workingmemory serves as state to represent the objects or variables currentlyunder consideration. This generally takes the form of fixed-points forsome function ƒ in the form x=f (x) where this equation has multiplestable values. For example, if

( ) is the “not” or inverse Boolean operation, an SR flip-flopimplements the recurrent relation x=

(

(x)), which has two stable fixed points: zero and one. As a practicalmatter, to maintain state indefinitely in actual implementations, f mustalso provide some amplification in order to overcome the effects ofdiffusion.

The Ψ-MAP associations, like the recurrence relation for the RSflip-flop, can be divided into a pair of mutually recurrent functions ofthe form y=f (x) and x=g(y), which implies x=g(f(x)). In the case of aflip-flop, x and y are Boolean variables, while in the case of a Ψ-MAPthey are functions defined on a manifold. These complementary orreciprocal relations statically retain their state and create ashort-term working memory. In an SR flip-flop, the functions f and gcorrespond to NAND (inverter) gates, while in a Ψ-MAP they correspond toΛ-MAPs.

The stable fixed points often correspond to the corners of a hypercubewhere the component values have been saturated or clipped. For example,a flip-flop can be viewed as a one-dimensional hypercube. This contrastswith the Ψ-MAP design where the stable states correspond to grayscale(non-saturated) images.

The Ψ-MAP design includes both a classification and an association.Initially, each Ψ-MAP goes through a classification phase where theinitial inputs progress along a trajectory to a stable fixed point.However, once the fixed point is reached, the mutually recurrentrelationships define an explicit and static association between x and y.In this respect, they are similar to bidirectional associative memories.

Bidirectional Associative Memories (BAMs), first described by Kosko(1988), explicitly define a mutually recurrent relationship. Subsequentresearch has further developed BAM implementations of associativememories. Shen and Balakrishnan (1998) used two mutually recurrentneural networks in a control system application. All of these networksrequire global connectivity, that is, each output is a function of eachof the inputs.

The BAM image association methods require global connections to theentire input image; each output pixel directly depends on all the inputpixels. Consequently, they suffer from low capacity and poor performancefor high-resolution images. The Ψ-MAP algorithms by contrast use mutualrecursion to form global associations with processing elements that haveonly local connections.

Cellular neural networks, which only have local connectivity, have alsobeen used to create (classification-type) associative memories. However,these associations are also defined as the correspondence between thebeginning and ending points of a path in a dynamical system and do notcontain static relationships corresponding to explicit associationsbetween two mutually recurrent state variables.

In contrast to many neural networks that use binary data values, theinventive model uses real values. Moreover, the inputs and outputs arereal-valued data defined on a continuum rather than on a discrete set ofpoints.

Neural field models have been studied for over 50 years. This approachmodels the behavior of a large number of neurons by taking the continuumlimit of discrete neural networks where the continuous state variablesare a function in space representing the mean firing rates. The primarydistinction between the prior art models and the model of the presentinvention is the physical quantity being measured. Neural field modelsattempt to describe the mean neuron firing rates as continuous quantity,while the Ψ-MAP array model describes the concentration ofneurotransmitters in the extracellular space as a continuous quantity.The localization of neurotransmitters to the space within the synapticcleft is an evolutionary optimization. In contrast with the neural fieldmodels, individual neurons in the Ψ-MAP array model correspond to pointssampled from the space on functions defined over the concentration ofneurotransmitters in physical space. Since neurons are samples from thisinfinite-dimensional space of functions, but are physically located inthree-dimensional space, their output values will appear discontinuous.

The manifold backpropagation algorithm (Puechmorel, Ibnkahla &Castianié, 1994) generalizes the vector backpropagation algorithm forneural networks (Ibnkahla, Puechmorel & Castanié, 1994) which is itselfa generalization of the common PDP backpropagation method. However, allof these algorithms take as inputs a finite set of points and aretherefore discrete. In the case of the multi-layer manifold neuralnetwork, the inputs, hidden layers and outputs are a finite set ofpoints located on a manifold. Each point has a matrix of adjustableweights representing a linear transformation in the tangent bundle. Thiscontrasts with the Ψ-MAP model where the data is continuous, inparticular a function representing the concentration of neurotransmitterpresent in the extracellular space. Furthermore, the weights are not afinite set of matrix coefficients but rather functions or measuresdefined in three-dimensional space.

Neural networks are functions that map input vectors to output vectorsbased on the values of a collection of real-valued weights. The manifoldthat results from viewing these functions as a space parameterized bythe weights is sometimes called a neural manifold (Amari, 1997;Santiago, 2005). For purposes of the present invention, we extend thisterm to the case where the number of weights is infinite, that is, theweights represent a function describing the sensitivity of the dendritictree or the output of the axonal tree in three-dimensional space.

Under reasonable assumptions about noise, functions defined on higherdimensional spaces are capable of encoding more information that thosedefined on lower dimensional spaces. Consequently, dimensional reductiontechniques result in an irretrievable loss of information. Thesetechniques are inherent in the concept that the CNS has developed sometype of feature extraction mechanism that converts functions defined oncontinua into tokens.

In the Ψ-MAP algorithms, there is no need to extract features or removethe image's intrinsic topological structure, which is preserved duringthe spectral transformations. The associations take place directlybetween the input image pixels and output image pixels. Consequently,there is no intervening space of lower dimension that forces a loss ofinformation. In contrast, indirect methods reduce image data to anintermediate form by finding discrete tokens or lower dimensionalfeatures such as edges and corners. These methods form associations byanalyzing features in the lower dimensional space.

Except for the initial low-level processing stages, many computer visionalgorithms are primarily concerned with segmentation, featureextraction, feature detection, and high-level processing. These areindirect methods that require the reduction of multi-dimensional imagedata to a lower dimensional space. The direct method of the presentinvention also contrasts in this regard from previous work onself-organizing maps, competitive learning and the reduction oftopographic maps to discrete vectors.

An “isomorphism” is a one-to-one mapping between two sets where themapping preserves analogous operations within each set. The Ψ-MAP designuses isomorphisms between functions and their spectral representationsin order to determine associations faster and more accurately.

Wavelet networks combine the functional decomposition of wavelets withthe parameter estimation and learning capacity of neural networks (Zhang& Benveniste, 1992; Iyengar, Cho, & Phoha, 2002). These algorithms learnthe wavelet coefficients and the parameters associated with scale andposition by minimizing an error term. Wavenets leave the scale andposition parameters fixed, and learn only the wavelet coefficients(Thuillard, 2002). These models are well suited for function learningand the approximation of arbitrary functions defined on

^(n) (Rao & Kumthekar, 1994; Zhang, Walter, Miao & Lee, 1995; Ciuca &Ware, 1997). The inclusion of a linear term reduces the number ofwavelets required (Galvão, Becerra, Calado & Silva, 2004). The estimatedcoefficients, which are the outputs of a wavelet network, can be used asinputs to a traditional neural network and used for classification andsystem identification (Szu, Telfer & Garcia, 1996; Deqiang, Zelin &Shabai, 2004).

There are several major differences between wavelet networks and theinventive model. First, the objectives are different: while waveletnetworks attempt to find wavelet coefficients that approximate afunction, the Ψ-MAP algorithms attempt to create an associative memorybetween functions by mapping the wavelet coefficients of the input imageto the wavelet coefficients of the output image.

Second, wavelet networks are defined in terms of a set of functions thatform a wavelet basis, while the methods described herein are defined interms of the continuous wavelet transform (CWT) or a wavelet frame thatcontains redundant information. This permits the creation ofassociations with fewer non-zero coefficients, and allows the use of thereproducing kernel to reduce noise and create stability in the recursiveconnections.

Since distinct pairs of functions in a wavelet basis are by definitionorthogonal, the value of the reproducing kernel will be zero at pointscorresponding to the pair. Consequently, a projection based on thereproducing kernel will have no effect, since any possible combinationof coefficient values corresponds to a valid image. As a result, whenusing a wavelet basis, the redundancy present in a CWT or a waveletframe representation cannot be used to reduce error and create stableminima. The linear dependence of the functions allows the use of thereproducing kernel provided there is sufficient redundancy.

Many neural models are stochastic and involve one or more randomvariables. These often take the form of a hidden Markov model. While anyanalog or continuous computation involves some amount of noise, theactual behavior of individual neurons does not appear to be random. Thefact that neighboring neurons usually exhibit completely differentresponse patterns is sometimes seen as evidence of an underlyingstochastic mechanism. However, as we will show, the spatialdiscontinuity in neuron firing rates is the result of sparsely samplingan infinite-dimensional space of functions.

In contrast with these methods, the Ψ-MAP model is entirelydeterministic. That is, given the same inputs it will predictablyproduce the same outputs.

In general, we will use lowercase letters for functions and variablesand uppercase letters for manifolds. Bold lowercase letters denotevector variables while bold uppercase letters denote vector spaces.

Script letters (e.g., F) designate transforms that map functions definedin one space to functions defined in another space. Brackets “[ ]” areused along with parentheses to specify the transformation of a functionas well as its arguments. For example the Fourier transform of afunction g(x) will be denoted by F[g](ω) where the argument ω is theradial frequency.

We will use the term “measure” to mean a signed measure, that is onethat can map sets to negative values (Kolmogorov & Fomin, 1970). Anintegral using a signed measure can be regarded as a shorthand notationfor an integral that is evaluated as the difference of two integralsusing ordinary, nonnegative measures.

The standard notation L^(p)(H,μ) refers to the space of functions on Hwhere the integration is done with respect to the measure μ (Royden,1988; Taylor & Lay, 1980). Functions actually refer to equivalenceclasses and consequently, two functions are equal when they are equalalmost everywhere, that is, except on a set of measure zero. Forsimplicity, we will usually drop the explicit specification of themeasure and denote the function spaces as L²(H) even when it is possibleto generalize the results to L^(p) (H) for p≠2. To make it easier toidentify functions and spaces, we often use the same letter in bothlowercase and uppercase to denote a function and the space on which itis defined, for example hεL²(H).

In the world of everyday experience, humans regard symbols as graphicalicons (Jung, 1964), visual representations that can be photographed andprinted on a sheet of paper. Computers on the other hand regard symbolsas the bit patterns contained in a single byte or word. When required,the glyph or a textual description of these efficient bitrepresentations is available from a table of character codes or adictionary. Physically and mathematically, these two types of symbolrepresentations are fundamentally irreconcilable. A photograph records ascalar or vector valued function defined on a rectangular region ofspace, that is, a two-dimension manifold, while a bit pattern identifiesa single element in a finite set of discrete points. One of the mostsalient characteristics of the Ψ-MAP algorithms is their ability tolearn and execute general-purpose operations on high-resolution images,without ever having to reduce these symbols to a bit pattern that fitsinto a single word of computer memory.

Just as a SR flip-flop is composed of two stateless NAND gates, eachΨ-MAP is composed of two stateless Λ-MAPs. These are labeled theexternal and internal Λ-MAPs analogous to the external and internallamina of the neocortex. In a steady-state condition, the outputs ofthese two Λ-MAPs form a reciprocal image pair.

The cerebral cortex is composed of approximately fifty Brodmann areas.The outputs of each Brodmann area, which arise primarily from theexternal and internal pyramidal layers, project to the inputs to one ormore other Brodmann areas. Each Brodmann area is analogous to a singleΨ-MAP, and we refer to an analogous collection of interconnected Ψ-MAPsas a Ψ-MAP Array.

Image masks, presumably originating from the thalamus, control theoperation of the array by specifying whether the individual Ψ-MAPsshould accept new inputs or retain their current contents. By overlayingthe content images, they can also focus attention onto specific regionsof interest.

While neural field models attempt to describe the mean firing rate ofneurons as a continuum, we instead model the concentration ofneurotransmitters in the extracellular space as a continuum. Modelingthe density of various chemical substances in space is straightforwardand commonplace. While neurotransmitters are generally confined to thesmall gap in the synaptic cleft, this can be seen as an evolutionaryadaptation to the problems of inertia and diffusion that would result iflarge quantities of neurotransmitters were present throughout the entireextracellular space. We describe a model where the release ofneurotransmitters by the axonal tree and the sensitivity of thedendritic tree, are both characterized by functions on thetwo-dimensional cellular surfaces embedded in three-dimensional space.In this model, information is not encoded in the neuron firing rates,but rather in the neurotransmitter concentrations.

Topologically, a network is a type of graph, described by graph theoryand the laws of discrete mathematics. Consequently, the intrinsiccharacteristics of a neural network are discrete. In order to move fromthe discrete to the continuous, we need to generalize the discreteconcepts of neural networks to continuous concepts of computationsdefined on manifolds. In effect, we need to replace summations withintegrals. Discrete and continuous analogs are common in science andengineering. For example, the continuous Fourier transform and thediscrete FFT, or the continuous and discrete wavelet transforms. Sometransforms, such as the Z transform are continuous in one domain anddiscrete in another. We refer to the continuous generalization of adiscrete neural network as a neural manifold (NM).

The standard projection neural network calculates the inner-productsummation of its input nodes with a vector of fixed weights. Since theoperations in a neural manifold are defined using Lebesgue integrals,the operands can contain Dirac delta (impulse) functions. We will showhow this mathematical formulation allows the neural manifold to subsumecompletely the functionality of the standard neural network. That is,for every projection neural network there is a corresponding neuralmanifold analog that performs an identical calculation. Furthermore, theLebesgue integral allows us to mix together any combination of discreteand continuous data. We demonstrate how a realistic physical structurerepresenting an actual neuron naturally leads to a neural manifoldmodel.

While the recursion allows neurons with local connections to form globalassociations, a mixture of region sizes from which the neurons makeconnections allows a Ψ-MAP to find the matching image more readily. Wecan extend this idea by “pre-computing” the average of small imageregions with various sizes and using these averages as inputs to theneurons. Rather than simply using averages, we further extend this ideaby using local spectral functions such as two-dimensional wavelets.These spectral functions can be identical to the well-known receptivefields in the retina of the eye. A topographic mapping, from the size ofthe receptive field in the retina, to the location within the thicknessof a single lamina in Brodmann area 17, the primary visual cortex, iswell documented. In the section on spectral transforms, we discuss howrather than being an unusual anomaly, that is unique to the visualsystem, the receptive fields of the retina are an expression of ageneral principal of operation that governs all of the Ψ-MAPs throughoutthe cortex.

Many spectral operators such as the wavelet transform, map functionsfrom a lower dimensional space to a higher dimensional space. The resultis that an arbitrary function in the higher dimensional space does notnecessarily correspond to a function in the lower dimensional space, andconsequently its inverse transform does not exist. The reproducingkernel allows us to define an integral operator, which is linear andorthogonal and maps an arbitrary function in the higher dimensionalspace to the nearest function for which the inverse transform exists.Performing this projection at the final stage in both Λ-MAPs, reducesvarious computational errors and guarantees that the Ψ-MAP outputsalways represent valid images.

Wavelet transforms and neural manifolds provide a framework in which wecan specify a detailed Ψ-MAP design. For the purposes of explanation andanalysis, we describe a single reference prototype. However, there aremany possible variations to this design that arise from numerousimplementation tradeoffs and alternate design patterns. The first ofthese is a choice between synchronous and asynchronous architectures andwe argue that while an asynchronous design is possible, it may beproblematic. If the synchronous alternative is chosen, we can replacethe standard square wave clock used in digital circuit design, with atriangular wave analog clock. Using the clock phase information, it ispossible to implement progressive resolution algorithms that begin withhigh-frequency details and progressively incorporate lower-frequencyspectral information.

The primary outputs of the neocortex arise from the two pyramidallayers. Since these outputs must travel a relatively long distance, inorder to conserve metabolic energy, teleonomic arguments suggest theneed for accurate encoding algorithms that can represent images with avery few number of neurons. We describe a nonlinear encoding method thatachieves this goal and is consistent with the known lateral inhibitionin the pyramidal cells.

With the understanding that many architectural design variations arepossible, we describe a detailed reference Ψ-MAP prototype and themathematics that govern it. We deconstruct and analyze the Ψ-MAP anddescribe each of the constituent components, including the integrationof multiple input images, the general association neural manifolds, themasking and multiplexing operations and the orthogonal projections.

For many neural networks, learning is a simple matter of associatingoutputs with inputs. However, because of the additional complexity ofthe Ψ-MAP several different types of learning are possible. We identifysix distinct learning categories and briefly characterize each one.

The veracity of a scientific model is ultimately judged by how well itpredicts real world phenomena. The Ψ-MAP Array model has many featuresin common with known facts about the neocortex. In the section onneurological and psychological correlates, we present a series of verybrief connections or links between established neuroscientific facts andcharacteristics of the Ψ-MAP Array model. Many of these deserve detailedstudy in their own right and the few sentences provided serve only toidentify the connection.

We can interpret an arbitrary collection of photographs as a set ofsymbols. If we wish to represent numbers or letters, we can pick a fontand record images of their alphanumeric glyph. Similarly, fixed-timesnapshots of any two-dimensional computational map may represent symbolsof external stimuli or motor control actions. Words can be representedby recording their acoustical sound and storing two-dimensionaltime/frequency spectrographs.

If a folder on a workstation contains a number of digital photographicimages, it is easy to imagine a black box that takes as input, acharacter string (the filename) and produces the associated photographas an output. Abstractly this simple program is an associative memorydevice that associates a string address and a photographic image.Moreover, the association is arbitrary in that we can choose anyfilename we want for any image. Now instead of a string address assumethat we use an image as an address. This new black box takes aphotograph as its input address and associates with it an arbitraryoutput photograph. This abstract black box represents a two-dimensionalmanifold association processor (2D MAP)

The black box described above took as an address a single input image,and produced as a result a single output image. However, a MAP can takeas input any number of image addresses and produce as output any numberof image results. Illustrated in FIG. 1, we show a two-input,single-output 2D MAP. The output of a multi-input MAP depends on all ofits inputs. That is, the output image may change when any of the inputimages changes. This abstract MAP model is stateless. It does notremember past values, and its output image depends only on the currentvalues of its input images. We will refer to a stateless, single-outputManifold Association Processor as a Λ-MAP (A from the Greek wordLogikos).

If images can represent symbols, then a two-input, single-output Λ-MAPcan function as an Arithmetic Logic Unit (ALU). Imagine ten photographicimages of the digits zero through nine. By learning two hundred imageassociations, the Λ-MAP can memorize the addition and multiplicationtables. If two images are used to represent “Condition X is True/False”,then any binary Boolean operation can be learned by memorizing fourimage associations. Thus, a Λ-MAP is logically complete andconsequently, with additional memory and control components, it cansimulate the operation of any digital computer.

The abstract Logic Manifold Association Processor or Λ-MAP of FIG. 1produces an output image that is associated with its input images. It isstateless since its current output does not depend on the previousinputs.

A modern digital computer typically contains both dynamic and staticRandom Access Memory (RAM). Static RAM or SRAM does not need to berefreshed and will maintain its state as long as power is supplied.Internally, inside each bit of SRAM, is a logic circuit equivalent tothe inverter loop shown in FIG. 2 a. This circuit contains twoinverters, each of which generates an output that is the logicalcomplement of its input. This circuit has only two stable states, onewhere the value of Q equals one and its compliment Q′ is zero, and theother where Q equals zero and Q′ is one. Other combinations of valuesfor Q/Q′ are unstable. If the circuit is ever momentarily forced into astate where Q and Q′ are both equal and then released, it will oscillatefor a few nanoseconds before it settles into one of the two stablestates.

FIG. 2 b shows the logic diagram of a standard S-R flip-flop designedusing two NAND gates. When the “set” (S′) and “reset” (R′) inputs areboth one, the circuit is equivalent to the inverter ring shown in FIG. 2a and will maintain its value indefinitely. If the S′ or R′ inputsmomentarily go to zero the flip-flop will set (Q=1) or reset (Q=0) andremain there until either the S′ or R′ input is changed again.

FIGS. 2 a and 2 b illustrate the fundamental unit of memory, a singlebit of storage, that can be constructed from two logic gates by creatinga circuit with a feedback loop. FIG. 2 a illustrates two inverters in aring “lock in place” a single bit and its complement. FIG. 2 b showsthat when the set (S′) and reset (R′) inputs are both one, the SRflip-flop is logically equivalent to the inverter ring. A zero appliedto either the S′ or R′ inputs can be used to change the state of thestored bit.

Now suppose we replace the single bits S′, R′, Q and Q′ in the S-Rflip-flop with two-dimensional images and we replace the two NAND gateswith two Λ-MAPs. This Set-Reset Manifold Association Processor (SR-MAP)is shown in FIG. 3, where we have relabeled the Q′ output as P.

Referring to FIG. 3, let Λ_(E)(S, P)=Q denote the external Λ-MAP in thefigure and let Λ_(I)(Q, R)=P denote the internal Λ-MAP. Let Null denotea predefined “blank” image that is identically equal to zero and let{(a₁, b₁), (a₂, b₂) . . . (a_(i), b_(i)) . . . (a_(n), b_(n))} be anarbitrary collection of n image pairs. Suppose we program Λ_(E) suchthat Λ_(E) (Null, b_(i))=a_(i) and program Λ_(I) such that Λ_(I)(a_(i),Null)=b_(i) for all i. Then when the R and S inputs are Null, the SR-MAPwill have n stable states corresponding to the n image pairs (a_(i),b_(i)). Consequently, we refer to the images that form an image pair(a_(i), b_(i)) as reciprocal images.

In addition to the above, suppose we have n input images (s₁,s₂,s₃ . . .s_(n)) and we add the additional associations to Λ_(E) such thatΛ_(E)(s_(i), X)=a_(i) for any image X. Then by changing the S input fromNull to we can force the SR-MAP from whatever state it is currently into the state identified by the image pair (a_(i), b_(i)). If the S inputnow returns to Null, the SR-MAP will remain in this state until eitherthe S or R input image changes again.

Illustrated in FIG. 3, the Set-Reset Manifold Association Processor orSR-MAP is analogous to the SR flip-flop. It comprises two Λ-MAPs labeledExternal and Internal. When Q=a_(i) and P=b_(i) these two reciprocalimages can “lock” each other in place until changed by the S or Rinputs.

Until now, we have been able to operate abstractly on manifolds assmooth continuous functions. In practice, these functions arediscretized, either on a regular grid such as the pixels in a digitalcamera or an irregular grid such as the rods and cones in the retina ofthe eye. We would like to construct the MAPs using neural networks.However, when every pixel in the input image is part of the calculationfor each neural network output pixel, for high-resolution images thenumber of inputs to the neural net becomes overwhelming and performancebegins to degrade. The same is true when the neural network training setcontains too many input/output pairs. Consequently, to construct a MAPwith a large number of high-resolution associations, a new type ofdesign is required.

FIG. 4 illustrates how Processing Elements with local support take theirreal-valued inputs from small regions in multiple input images andproduce a single real-valued output corresponding to a single pixel. Theregion of support may be narrow as shown for image A or broad as shownfor image B. A lattice of PEs operating in parallel is used to generatethe entire output image.

For the purpose of analysis, we partition the overall neural network anddefine an abstract Processing Element (PE). Each PE(i,j) accepts a smallset (vector) of real-valued inputs and produces a single real-valuedoutput corresponding to the pixel (i,j) in the output image. We will usethe term lattice to describe a collection of PEs which form an entireoutput image. Each PE(i,j) has in, input connections where m_(i,j) ismuch smaller than the total number of pixels in an input image. Such anetwork partition is illustrated in FIG. 4. Note that when an image issampled at a particular location, we assume that derivatives of anyorder desired are also available at that location.

The particular set of m_(i,j) inputs chosen could be based on atwo-dimensional Density Function (DF) similar to a 2D probabilitydensity function (Gnedenko, 1968; DeGroot and Schervish, 2001). Pixelsin the input image where the DF is relatively larger are more likely tobe chosen as inputs to a particular PE. Borrowing from the terminologyof real analysis, we define the support of a PE to be those points inthe image where the DF is non-zero. If the support of a PE is containedin a region of radius d, where d is small relative to the size of theimage, then we say the PE has local support.

We assume that the DF for each PE(i,j) shifts with the relative position(i,j) so that the spatial relationship between the output pixel and theinput DF remains approximately the same. Consequently, neighboring PEswould have overlapping regions of support. Note however that the designdoes not restrict the shape of the DF. FIG. 4 shows a PE with inputstaken from three images but each having different DFs. Even though thenumber of input samples taken from each image is the same, one supportregion is relatively “narrow”, taking samples that are in close physicalproximity, while another region is “broad”, taking samples that aresparsely distributed over a larger area. In general, the DFs need not besymmetric, or even centered around the location of the output pixel.

For a direct implementation of a Λ-MAP, the simple design strategydescribed above and illustrated in FIG. 4 will not work; the outputimage will consist of a melee of uncoordinated PE results. We need anarchitecture that will integrate the results of the individual PEs intoa coherent whole.

It is impossible to construct a simple feed-forward implementation of aΛ-MAP using neural networks with local support because there is no wayto coordinate the individual PE outputs to form a consistent globalimage association. It is however possible to construct an SR-MAP. Thisis because the two Λ-MAPs, which make up the SR-MAP, form an image loop.The output of any given PE feeds into the local support of several PEsin the opposite Λ-MAP. These in turn may form many local loops that feedback into the original PE. FIG. 5 illustrates how this can occur byshowing two connected PEs in the two Λ-MAPs.

We use the term Ψ-MAP (Ψ from the Greek word Psyche) to refer to aSet-Reset Manifold Association Processor that is constructed from PEswith local support.

The overall behavior of the Ψ-MAP is an emergent property of theinternal algorithms used in the PEs and how their DFs overlap andinteract. If the PEs are not carefully designed, then the associationprocessor as a whole will operate poorly or not at all. A primaryconcern is the stability of the Ψ-MAP. Take as an example the SR-MAPdescribed above that has been trained to remember a collection ofreciprocal image pairs (a_(i), b_(i)). If an input s_(i) forces theoutput of the external Λ-MAP to image which in turn forces the internalΛ-MAP output to b_(i), then when the S input returns to the null image,the Ψ-MAP must remain locked in the (a_(i), b_(i)) state. If themanifold processor is unstable, then even a small error in one of theoutput images may cause the outputs to drift away from the correctstate. It is therefore imperative that reciprocal image pairs correspondto local minima in the overall Ψ-MAP energy function causing outputs tomove toward stored recollections rather than away from them.

Illustrated in FIG. 1, the Ψ-MAP uses the image feedback loop in theSR-MAP to construct global associations from local connections. A singlerecurrence relation between two processing elements is illustrated wherethe output of each PE forms part of the other's input.

It is possible that a Ψ-MAP could oscillate for extended periods whilesearching for a solution. Even when forming a clear recollection, someamount of oscillation may occur as part of the normal associationprocess. It is also possible that the manifold processor may settle intofalse local minima where the output images represent “montages”comprising parts of several indistinct associations. These false localminima may be perceived as faint or “hazy” memories.

Because of the lamina of the neocortex, the previous discussion focusedon the association of images, that is, functions defined ontwo-dimensional manifolds. However, nothing limits us to two dimensions.The extension of the Λ-MAP, SR-MAP and Ψ-MAP concepts to n dimensions issimple and straightforward. In particular, the recognition ofthree-dimensional objects is critical to survival and it seems probablethat the CNS uses a 3D-MAP for this purpose rather than rely ontwo-dimensional projections. Additionally, one-dimensional Ψ-MAPs cananalyze functions defined in time or frequency.

It is tempting to try to create a zero-dimensional Ψ-MAP based on anarbitrary collection of symbols or words. The problem with this is thelack of a suitable metric. The ISO Latin character set is mapped to theintegers 0-255 which inherit a metric from the reals (∥x−y∥) butfractional values are meaningless and synonyms are not defined by closeproximity in a dictionary. Consequently, it is difficult or impossibleto establish a meaningful notion of convergence. In the Ψ-MAP Arraymodel, synonyms are also not similar to each other phonetically, buttheir spectrograms are associated directly or indirectly with otherimages which are close to each other in terms of a quantifiable metric.

If we assume that each Brodmann area in the neocortex corresponds to aunique Ψ-MAP, then the collection of all the Brodmann areas constitutesa Ψ-MAP Array as shown in FIG. 6.

Each line in the FIG. 6 corresponds to a two-dimensional image. Tosimplify the diagram, we use an Input/Output bus (“I/O Bus”) notationwhere the horizontal rows along the top of the diagram represent images,and the dots represent connections. The array transmits sensory inputsand motor control outputs as two-dimensional computational maps.Figuratively via the I/O bus, each Ψ-MAP generates and sends two outputimages corresponding to its two Λ-MAPs. However, a Ψ-MAP can have anynumber of input images. This is illustrated in FIG. 4 where we showedhow a neural network with local support could accept inputs from threeseparate images. Using topographic alignment, each Ψ-MAP can integratetogether any number of images from the I/O bus.

In the Ψ-MAP Array model of the neocortex as illustrated in FIG. 6, eachΨ-MAP, Ψ_(i), corresponds to a separate Brodmann Area. Lines in thediagram correspond to 2D images, and dots represent specific input andoutput connections.

Inside every CPU (Central Processing Unit), a control unit coordinatesthe flow of data between the various components. As an extreme example,one possible control structure for the Ψ-MAP Array would be to have allof the Ψ-MAPs always transmitting their outputs and continuallyreceiving and processing entire images from all of their inputs.However, a more dynamic and flexible design would allow some of theΨ-MAPs to be active while others remain inactive or quiescent. Theinputs received from the quiescent MAPs may correspond to “off” or nullimages. While a particular Ψ-MAP may have many inputs, only a subset(including the empty set) may be active at any given time. The MAPs mayalso choose to retain their contents or selectively accept new inputs.

Every area of the neocortex has reciprocal connections with thethalamus, which is known to play an important role in consciousness.Based on observed behavior and the types of real-world problems that theCNS must solve, an agile and flexible control mechanism must be able tofocus on subsets or regions. We can view these manifold regions as 2D“control masks” that overlay Ψ-MAPimages and regulate the Ψ-MAPprocessing and I/O. Based on the current images and newly formedassociations, each Ψ-MAPin the neocortex relays an “activity” mask tothe thalamus. The thalamus in turn relays a “process/hold” control maskback to the Ψ-MAP. The masks blanket the entire extent of the Ψ-MAPArray “content” images, and consequently they can identify an arbitraryregion within the image. The portions of the MAP images inside themasked region are active while those outside of the region arequiescent. In this way, the thalamus can focus attention on the entireimage or on some very small detail. Given the variety of computationalmaps, control masks can direct our attention to a small region in thevisual field, a single sound in a word, a particular part of the bodysurface, or the motion of a single muscle. Using time varying controlmasks, we can spatially scan a visual image, or temporally scan throughthe sounds of a sentence recorded in a spectrogram.

The incorporation of masks into the control structure of the Ψ-MAP arraysignificantly extends its functional capacity. With control masks, inaddition to the numerous image combinations, we can also createassociations based on various selected parts of the content images.Thus, the control masks direct how and when the associations are formed.

FIG. 7 illustrates how the Fourier transform maps functions of the realvariable x to the spectral space of functions of the frequency variableω. The two spaces are isomorphic under the convolution andmultiplication operators. An isomorphism is a one-to-one mapping betweentwo sets, where the result of an operation on elements of one set, mapsto the result of a second operation performed on the mapped elements.

It often happens that operations that are difficult to perform in onespace can be transformed to another space where the computation iseasier. An example of this is illustrated in FIG. 7 where theconvolution of two functions ƒ and g is computed by taking their FourierTransforms, multiplying the results and then taking the inverse FourierTransform. Since the Fast Fourier Transform (FFT) can be computed intime O(n log n) and a convolution requires time O(n²), for large n theindirect approach using forward and inverse FFTs is quicker. Othertransform methods including the cosine transform, the Walsh transform,or the wavelet transform can also be used to establish isomorphicmappings between function spaces.

We will explore a similar isomorphism between the space of images andtheir spectral representations. However, instead of convolution andmultiplication, the operation in both spaces will be the establishmentof an association relationship between two arbitrary functions.

Axons from the on and off-center ganglion cells in the retina form theoptic nerve. These cells receive inputs from nearby photosensitive rodsor cones in a roughly circular receptive field that contains both acenter area and a surrounding ring. For the on-center ganglion cells,the maximum excitation is achieved when the center region is illuminatedand the surrounding ring is dark, while just the opposite is true for anoff-center ganglion cell. Different ganglion cells have receptive fieldsthat vary in size; the M (for magni, or large) cells have receptivefields that cover a wide area while the P (for parvi, or small) cellshave relatively narrow receptive fields. It has been noted (Kelly 1975;Nevatia, 1982; Horn & Brooks, 1989) that these receptive fields aresimilar to spherically symmetric functions where the difference in thesize receptive fields of the M and P cells is given by the scalingparameter s.

The axons of the P and M ganglion cells remain segregated as theyproject onto the lateral geniculate nucleus (LGN). The ventral layers ofthe LGN, known as the magnocellular layers receive their inputs mainlyfrom the M ganglion cells while the dorsal parvocellular layers receiveinputs mainly from the P ganglion cells. Moreover, as the neurons in theLGN project to the primary visual cortex (Brodmann area 17), theymaintain this separation. Indirectly though the LGN, the majority of theM ganglion cells in the retina, map to layer 4Cα in the cortex, whilethe P ganglion cells map to layer 4Cβ directly beneath it. This impliesthat if we establish a local (x,y,z) coordinate system in the primaryvisual cortex with the xy-plane parallel to the layers of the cortex andthe z-axis perpendicular to the surface, the size or scale of thereceptive field will be topographically mapped to the z dimension.

Throughout the cerebral cortex, the grey matter is plainly visible on amacroscopic scale. That is, the layers which make up the cortex have amulti-cellular thickness in the perpendicular z direction that gives thelayers a three dimensional structure. A central thesis behind theinventive method is that the topographic mapping of receptive fields ofincreasing size map to the third (z) dimension is not an anomaly uniqueto visual processing, but rather, is a general principle that underliesall computations in the cerebral cortex.

Like the Fourier transform, the wavelet transform maps functions to anisomorphic frequency space where some computations can be performed moreaccurately with fewer operations. However, unlike the complexexponential function exp(iωx) in the Fourier transform, which stretchesto plus and minus infinity, wavelets are concentrated at a particularpoint in time or space. Moreover, there is not one single, unique set ofwavelets, but an infinite number of wavelet sets that are defined onlyin terms of a few general characteristics. In one-dimension, a singlemain wavelet φ, which is normalized and symmetric about the origin, cangenerate a family of wavelets at position x and scale s (Mallat, 1999)

$\begin{matrix}{{\phi_{x,s}(\xi)} = {\frac{1}{\sqrt{s}}{\phi \left( \frac{\xi - x}{s} \right)}}} & (1)\end{matrix}$

We will refer to wavelets with a relatively large value of scalingparameter sεR₊ as low-frequency wavelets and those with a relativelysmall value of s as high-frequency wavelets. The wavelet transform{tilde over (f)} of a function ƒ is given by

$\begin{matrix}\begin{matrix}{{\overset{\sim}{f}\left( {x,s} \right)} = {{W\lbrack f\rbrack}\left( {x,s} \right)}} \\{= {\int_{- \infty}^{+ \infty}{{f(\xi)}{\phi_{x,s}^{*}(\xi)}\ {\xi}}}} \\{= {\langle{f,\phi_{x,s}}\rangle}}\end{matrix} & (2)\end{matrix}$

where φ* denotes the complex conjugate of φ. The inverse wavelettransform completes the isomorphism illustrated in FIG. 7 by providing atransform of the computed results back to the original space.

$\begin{matrix}\begin{matrix}{{f(\xi)} = {{W^{- 1}\left\lbrack \overset{\sim}{f} \right\rbrack}(\xi)}} \\{= {\frac{1}{C_{\phi}}{\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{{\overset{\sim}{f}\left( {x,s} \right)}{\phi_{x,s}(\xi)}\ {x}\frac{s}{s^{2}}}}}}}\end{matrix} & (3)\end{matrix}$

The constant C_(φ) is given by

$\begin{matrix}{C_{\phi} = {\int_{0}^{+ \infty}{\frac{{{\hat{\phi}(\omega)}}^{2}}{\omega}\ {\omega}}}} & (4)\end{matrix}$

where {circumflex over (φ)} is the Fourier transform of φ.

Several methods are available that can generate sets of multidimensionalwavelets whose linear combinations are dense in L²(R^(n)).

Let the vector x=(x₁, x₂, . . . , x_(n))εR^(n) denote the waveletposition, the vector s=(s₁, s₂, . . . , s_(n))εR₊ ^(n) the set ofscaling factors, and ξ=(ξ₁, ξ₂, . . . , ξ_(n))εR^(n) the dummy variablesof integration. A straightforward multidimensional extension of thewavelets specified in (1) is to let m=n and form the separablefunctions:

$\begin{matrix}{{\phi_{x,s}(\xi)} = {\frac{1}{\sqrt{s_{1}s_{2}\mspace{14mu} \ldots \mspace{14mu} s_{n}}}{\phi \left( \frac{\xi_{1} - x_{1}}{s_{1}} \right)}{\phi \left( \frac{\xi_{2} - x_{2}}{s_{2}} \right)}\mspace{14mu} \ldots \mspace{14mu} {\phi \left( \frac{\xi_{n} - x_{n}}{s_{n}} \right)}}} & (5)\end{matrix}$

The parameter space for these wavelets, (x,s)εR^(n)×R₊ ^(n), hasdimension 2 n, and the wavelets mix information at many differentscales, s_(i),s_(j).

Separable multiresolution methods can be used to construct separablewavelets that have the same scale parameters (Mallat, 1999). Theresulting multiresolution wavelets have a parameter space,(x,s)εR^(n)×R₊, of dimension n+1.

Of particular interest in neurobiology are spherically symmetricwavelets, which can be expressed in the form ω(x)=f(∥x∥); xεR^(n) forsome one-dimensional function ƒ. The scale parameter for sphericallysymmetric wavelets is a single real-valued positive number sεR₊.Consequently, the overall parameter space has dimension n+1.

For two-dimensional images, an example of a spherically symmetricwavelet is the Mexican hat wavelet, which is somewhat similar to thereceptive fields in the retina. However, because it is the normalizedsecond derivative of a Gaussian function, it has non-zero values out toinfinity. This implies that to compute such a wavelet, many of theneuron dendritic trees would need to extend over the entire range of theimage, thereby reducing some of the advantages of local support. Forthis reason, we are mainly interested in wavelets with compact support(Daubechies, 1992), in particular wavelets that are zero outside a smalllocal region.

If a wavelet is spherically symmetric, so is its Fourier transform.Thus, {circumflex over (φ)}(ω)=)χ(∥ω∥) for some function χ, and theadmissibility condition is

$\begin{matrix}{C_{\chi} = {{\int_{0}^{+ \infty}{\frac{{{\chi (\omega)}}^{2}}{\omega}\ {\omega}}} < \infty}} & (6)\end{matrix}$

For fεL²(R^(n)) the wavelet transform, {tilde over (f)}, is defined byextending the integration in Equation (2) to n dimensions. The inversetransform inn dimensions (Daubechies, 1992; Addison, 2002) is given by

$\begin{matrix}\begin{matrix}{{f(\xi)} = {{W^{- 1}\left\lbrack \overset{\sim}{f} \right\rbrack}(\xi)}} \\{= {\frac{1}{C_{\chi}}{\int_{R_{+}}{\int_{R^{n}}{{\overset{\sim}{f}\left( {x,s} \right)}{\phi_{x,s}(\xi)}\ {x}\frac{s}{s^{n + 1}}}}}}}\end{matrix} & (7)\end{matrix}$

So far, we have discussed wavelets defined on R^(n). However, the CNSprocesses many different types of the topological manifolds (Greer,2003). These include the surface of a sphere, disks, the surface of acylindrical section, and even kinetic images composed of the union ofdisjoint muscle cross-sections. The common thread running through thesespaces is that they are mathematical manifolds, that is, around eachpoint there is a neighborhood that is diffeomorphic to an open set inR^(n). We will use M to denote an arbitrary n-dimensional manifold.Wavelets on spherical surfaces are widely used in science and spectraloperators on any manifold can be defined using linear integraltransforms.

Separable wavelets with multiple asymmetric scaling factors,multiresolution wavelets and spherically symmetric wavelets defined on Mall have a scaling parameter space R where 1≦m≦n. All of these waveletsare constructed by forming dilated “daughter” wavelets from a singlemain wavelet. The amount of dilation is specified by the scalingparameter s as in Equation (1) or the {s_(i)} in Equation (5).

While the receptive fields of the retina resemble two-dimensionalwavelets, there is evidence to suggest that they are not simply dilatedversions of one another. The response characteristics of the largerlow-frequency receptive fields have a dissimilar shape and contain adifferent number of periodic cycles than the smaller high-frequencyreceptive fields. Therefore, it is impossible to make these spectralfunctions congruent with each other by a simple dilation. Thesefunctions are not dilated versions of a single main wavelet, but theymay form a frame (Mallat, 1999). For any position xεM and scaling vectorsεR^(m) we can define a set of functions {φ_(x,s)} that have local spaceand frequency characteristics. We can view these as a generalization ofwavelets similar to a windowed Fourier transforms and where the windowis allowed to contract and change shape with increasing frequency. Whilethese functions may not be scaled versions of a single main wavelet,they have wavelet-like spectral characteristics and we can match themexactly with the receptive field characteristics of the retina. We let Sdenote the generalization of the continuous wavelet transform and defineit in terms of the inner product of the wavelet-like functions on M.

$\begin{matrix}\begin{matrix}{{{\lbrack f\rbrack}\left( {x,s} \right)} = {\langle{f,\phi_{x,s}}\rangle}} \\{= {\int_{M}{{f(\xi)}{\phi_{x,s}^{*}(\xi)}\ {\xi}}}}\end{matrix} & (8)\end{matrix}$

The spectral transform S is qualitatively similar to the wavelettransform W in its ability to provide a range of frequency informationat a localized position in space. A question that may arise is how toform an orthonormal basis for such a general set of wavelet likefunctions. In the following analysis however, a set of basis functionsis not required. What is required is a spectral transform that has someof the key properties of the continuous wavelet transform. Since acountable set of functions that form a frame or an orthonormal basis isnot necessary, it is important to list what properties are actuallyrequired.

For a general spectral transform 5, we will assume the following list ofproperties.

-   -   (a) We assume S transforms functions defined on an n-dimensional        manifold M, and is defined by Equation (8) for some set of        spectral functions {φ_(x,s)(ξ)}, where xεM is the “center”        location, and the vector sεR^(m) denotes the scaling factors. We        will refer to the n+m dimensional parameter space J=(M×R^(m)) as        a spectral manifold.    -   (b) We require that a reconstruction formula corresponding to S,        exist and be well defined. Let U be the subspace of L²(J) that        corresponds to transforms S[f] or some set of functions. For        {tilde over (f)}εU we denote the inverse transform as S⁻¹[{tilde        over (f)}] and require that {tilde over (f)}=S⁻¹[S[f]] almost        everywhere.    -   (c) For continuous wavelet transforms, the reproducing kernel        described below can be derived from the reconstruction formula.        For the case of a general spectral transform, we require that an        equivalent reproducing kernel exist, and that it can be used as        a kernel for a linear orthogonal projection onto U.

The extension to the Ψ-MAP computational model can be defined on anyn-dimensional manifold M with functions {φ_(x,s)} forming an n+mdimensional spectral manifold J. The functions {φ_(x,s)} could beseparable wavelets (with m=n), multiresolution wavelets, sphericallysymmetric wavelets (with m=1), other types of wavelets, or a general setof spectral functions that are not wavelets at all, but satisfy thethree postulates listed above.

Each of the Brodmann areas {Ψ_(i)} has a unique flat shape that projectsalong the direction of the cortical columns onto a two-dimensionalmanifold M_(i). We hypothesize that in addition to the retinalprojections onto the primary visual area in the neocortex, the otherBrodmann areas also use spectral functions that resemble sphericallysymmetric wavelets defined on two-dimension images. The spectralfunctions are parameterized by a center position (x,y)εM_(i) and asingle real-valued scale factor sεR₊. The resulting three-dimensionalmanifold J_(i)=(M_(i)×R₊) corresponds to a cortical layer of a singleBrodmann area. To simplify the diagrams, we will draw thetwo-dimensional manifolds {M_(i)} as rectangles with the understandingthat in the neocortex, they will have very irregular outlines and due tothe cortical sulci and gyri their actual physical shape will be veryconvoluted.

Examining the Ψ-MAP shown in FIG. 5, we now replace the images withthree-dimensional “slabs” of thickness z₀. The scaling parameter sε(0,∞)is monotonically mapped to the interval (0,z₀) that corresponds to thephysical thickness of a single cortical layer in a particular Brodmannarea. In the resulting three-dimensional spectral manifold, the pixelsshown in FIG. 5 now correspond to voxels that represent the magnitude ofspecific spectral functions. A processing element, PE, takes as inputsvoxels in a three-dimensional input spectral manifold, and produces as aresult, values in a three-dimensional output spectral manifold.

Low-frequency spectral functions measure components over a large area ofan image. Consequently, even though the PE has only local connectionsnear an image point (x₀,y₀), if the connections extend through theentire thickness of the cortical layer, its output value can changebased on changes in the input image from the finest to the coarsestlevels of detail. Moreover, the recursion in the Ψ-MAP allows the outputvalues of PEs that correspond to low-frequency spectral functions topropagate quickly throughout the entire MAP in the search for an overallglobal solution.

We have limited our discussion to Ψ-MAPs that can recall associationsbetween scalar value functions. We can however build Ψ-MAPs thatassociate complex or vector valued functions simply by using PEs thathave complex or vector valued inputs and outputs.

In the n+m dimensional case, we can smooth a function ƒεL²(R^(n)) usinga convolution kernel γ that is dilated by sεR₊ ^(m). The functions

$\begin{matrix}{{\phi_{i,x,s} = {- \frac{\partial\gamma}{\partial x_{i}}}};{1 \leq i \leq N}} & (9)\end{matrix}$

allow us to define a vector valued function

g(x,s)=(

f,φ _(1,x,s)

. . . ,

f,φ _(N,x,s)

)  (10)

that estimates the gradient of f at a scale s. The function g can beused to estimate the partial derivative of fin the direction of a unitvector n by taking the vector inner product n·g.

We can further enlarge the vector space to obtain estimates of higherorder derivatives of any degree required by taking higher orderderivatives of the convolution kernel γ. Expanding the range of theoriginal function, to a vector space of derivative values at multiplescales, facilitates the formation of Ψ-MAP image associations whereedges or gradations in value are important.

Insight into the general principles that govern the nervous systemrequires the analysis of nomological models at many levels of detailstretching from biochemistry to ethology. At the level of the Λ-MAP andthe Ψ-MAP Array, the operations are expressed in terms of associationsformed between functions defined on multidimensional manifolds. Amanifold is by definition a continuum. In contrast, a network is adirected graph, which by definition is discrete. A basic principleunderlying the use of neural networks is that any given mental state canbe described by a discrete N-dimensional vector of activation values.The unambiguous mathematical distinction between countable discrete setsand uncountable continua implies that, at this level of abstraction, theterminology of networks is inconsistent with the phenomena we areattempting to describe. Consequently, for mathematical correctness andclarity, we use the term neural manifold to denote the implementation ofa general mechanism that transforms functions defined on continuousdomains. More formally, for manifolds H and Q, and an arbitrarycollection of inputs {h_(β)}⊂L²(H) and associated outputs {q_(β)}⊂L²(Q),a neural manifold G, is defined as a physical realization of a member ofthe set of transforms {G_(α)}, for which q_(β)=G_(α)[h_(β)], ∀β. As wewill see, a neural manifold completely generalizes a neural network, andany calculation performed using a neural network has a simple and directanalog to a calculation performed using an NM.

When an urgent decision must be made, an organism that can effectivelyprocess clear and precise information has an advantage over one whoseinformation processing is dull and indistinct. Based on teleonomicarguments, we must assume that evolution will favor manifoldrepresentations that have a high effective resolution over those with alow resolution when other factors such as processing time, metabolicenergy and cellular hardware remain the same. Consequently, while thefast transmission of images must take place at the courser level ofaxons, the actual representation and processing takes place at the muchfiner level of neurotransmitters.

A neural manifold is an abstract model used to describe the density ofneurotransmitter molecules and the operation of neuron patterns found inthe CNS that, operating in parallel, transform functions defined oncontinua. It must be expected that many types of NMs have evolved overthe course of evolution to solve the specialized equations unique toaudition, vision and robotics. However, in the analysis of theneocortex, our primary concern is the general association NMs used torecall relationships between images.

At a finer level of detail, the cellular level, we begin to examine thecomputations of a single processing element. At this level ofabstraction, we use a model that has continuous operands but discreteoperators. The operands are functions representing the density of thecontinuous neurotransmitter “clouds”. However, the operators, that is,the neurons, are viewed as discrete computational units.

At next finer level of detail, the molecular level, both the operandsand operators are discrete. In the realm of natural computation, theneurotransmitters are now viewed as individual discrete molecules. Inthe realm of digital computers, at the molecular level, the operands arediscrete multidimensional arrays of numbers and the operators arediscrete arrays of CPUs.

FIG. 8 illustrates the realization of a NM at the cellular level. Inthis diagram, the input space H and the output space Q are boththree-dimensional spectral manifolds parameterized by (ξ,η,ζ) and(x,y,s) respectively. The PEs whose outputs pass through the boundarybetween H to Q are parameterized by the integer variable i.

FIG. 8 illustrates how the transformation can be implemented with anarray of processing elements {g_(i)}, where each element simulates theoperation of a single neuron. Each g_(i) is composed of a receptormeasure a cell-body-transfer function σ, a temporal filter ω, atransmitter function μ_(i) and cell-membrane-transfer functions χ_(d)and χ_(a). The cell-body-transfer function, σ, and thecell-membrane-transfer functions, χ_(d) and χ_(a), can be assumed totake a form similar to neural-network sigmoid activation functions andbe uniform throughout all cells, dendrites and axons. The receptormeasure μ_(i)(ξ,η,ζ) models the operation the dendritic arbor in theinput manifold H, while transmitter function τ_(i)(x,y,s) models thesignal distribution along the axon and the concomitant release ofneurotransmitters into the output manifold Q.

FIG. 8 illustrates the processing element g_(i) models the operation ofa single neuron. The receptor measure μ_(i) converts the continuousdistribution of neurotransmitter in the spectral manifold H to a singlereal value, while the transmitter function τ_(i), converts a single realvalue to a continuous distribution of neurotransmitter in the outputmanifold Q. The operation ω models the neuron's temporal response whileσ models the nonlinear response of the cell to the dendritic inputs. Thenonlinear response of the dendrite cell membrane is represented by χ_(d)and the nonlinear response of the axon cell membrane is represented byχ_(a).

The receptor measures {μ_(i)} can be visualized as the three-dimensionaldendritic tree corresponding to neuron i, where the dendrites have been“painted” with a shade of grey corresponding to their sensitivity to aparticular neurotransmitter. When multiplied by the actual concentrationof neurotransmitter present in the extracellular space, and integratedover a region of space that contains the dendritic tree, the result is afirst-order approximation of the neuron's receptor sensitivity.Mathematically, the {μ_(i)} are signed measures (Kolmogorov & Fomin,1970; Rudin, 1976) which define functionals that convert a function hdefined on H to a real value.

The dendritic tree computation is defined using Lebesgue integration as

$\begin{matrix}{{output} = {\int_{H}{{h\left( {\xi,\eta,\zeta} \right)}\ {{\mu_{i}\left( {\xi,\eta,\zeta} \right)}}}}} & (11)\end{matrix}$

To demonstrate why a neural manifold calculation subsumes thefunctionality of the standard neural network model, we examine thecomputation performed by a single-layer network with a single outputnode. For an n-dimensional input vector x=(x₁ . . . x_(n))^(T), a weightvector w=w₁, . . . , w_(n))^(T), and a transfer function σ, the output,y, of a typical neural network is given by

$\begin{matrix}{y = {{\sigma \left( {w^{T}x} \right)} = {\sigma \left( {\sum\limits_{k = 1}^{n}{w_{k}x_{k}}} \right)}}} & (12)\end{matrix}$

To construct an analogous NM, identify the input vector x with any setof n distinct points {(ξ_(k),η_(k),ζ_(k)); 1≦k≦n} in H, and let theinput vector values x_(k)=h(ξ_(k),η_(k),ζ_(k)) be defined by some inputfunction hεL²(H). Let {δ_(k)} be the set of three-dimensional Diracdelta functions (product measures) defined by

δ_(k)=δ(ξ−ξ_(k))δ(η−η_(k))δ(ζ−ζ_(k))  (13)

For a single PE, g_(i), assume the temporal filter ω has achieved asteady-state where the output is equal to its input, and let thetransfer function σ be the same as the one used for the neural network.Setting

$\begin{matrix}{\mu = {\sum\limits_{k = 1}^{n}{w_{k}\delta_{k}}}} & (14)\end{matrix}$

We have

$\begin{matrix}\begin{matrix}{{\sigma \left( {\int_{H}{h\ {\mu}}} \right)} = {\sigma \left( {\int_{H}{{h\left( {\xi,\eta,\zeta} \right)}\left( {\sum\limits_{k = 1}^{n}{w_{k}\ {\delta_{k}}}} \right)}} \right)}} \\{= {\sigma \left( {\sum\limits_{k = 1}^{n}{w_{k}\ {h\left( {\xi_{k},\eta_{k},\zeta_{k}} \right)}}} \right)}} \\{= y}\end{matrix} & (15)\end{matrix}$

Thus, a neural manifold PE with the measure μ, performs the samecalculation as the single-layer projection neural network. Thebiological meaning of the measure μ defined above is a mathematicalmodel of a neuron with n points (idealized synapses) each withsensitivity w_(k) and located at the spatial positions(ξ_(k),η_(k),ζ_(k)) inside the input manifold H. The measures {μ_(i)}allow us to model precisely the shape and sensitivity of the dendritictree for the neuron identified with each PE g_(i). The use of theLebesgue integral, instead of the conventional Riemann integral, allowsus to model neurons that are in effect able to discriminateneurotransmitter concentration at a single point (ξ_(k),η_(k),ζ_(k)),but at the same time, may also exhibit sensitivity over entire regions.

To realize the receptor measures on a digital computer, we convert thespectral manifold to a three-dimensional grid of floating-point values.At this point, the integral again becomes a summation. However, sincethe receptor measures are defined on a manifold, they are independent ofthe resolution of the grid. Consequently, we can evaluate them on ahigh-resolution grid and then down sample them to the equivalent weightsat a lower resolution for other computations.

The temporal filter ω can help control oscillations and add stability tothe overal Ψ-MAP. These may be analog filters in the case of continuoustime neural models or finite or infinite impulse response filters in thecase of discrete time digital implementations (Oppenhiem and Schafer,1975; Rabiner and Gold, 1975).

The relationship between the concentration of neurotransmitter in theextracellular space and the gating of the ion channels in the dendritictree is characterized by the transfer function χ_(d). At some point,increasing the concentration of neurotransmitter has a diminishingeffect on the ion channels; therefore, this function is nonlinear.Similarly, χ_(a) characterizes the nonlinear release ofneurotransmitters by the axons terminals as a function of the neuronfiring rate. The two cell-membrane-transfer functions, χ_(d) and χ_(a),as well as the cell-body-transfer function σ are analogous to a sigmoidtransfer function, such as 1/(1+exp(−x)) or the hyperbolic tangentfunction, that is used in neural networks. In the following, we willassume that χ_(d) and χ_(a), are real-valued functions of a single realvariable and are uniform over the entire cell membrane. The spatialvariations in the responses are represented using μ_(i) and τ_(i).

The transformation back to a continuous function q results from scalingeach transmitter functions τ_(i) by the output of the temporal filter.If we include the time variable t, the complete output q is given bysumming over all of the PEs

$\begin{matrix}{{q\left( {x,y,s,t} \right)} = {\sum\limits_{i}{\chi_{a}\begin{pmatrix}{\left( {{\sigma\left( \begin{matrix}{\int_{H}{\chi_{d}\left( {h\left( {\xi,\eta,\zeta,t} \right)} \right)}} \\{{\mu_{i}\left( {\xi,\eta,\zeta} \right)}}\end{matrix}\  \right)}*{r(t)}} \right) \cdot} \\{\tau_{i}\left( {x,y,s} \right)}\end{pmatrix}}}} & (16)\end{matrix}$

where * denotes the convolution operator and r(t) is the impulseresponse of temporal filter ω. The integrals with respect to themeasures μ_(i), and the summation over the transmitter functions τ_(i),in effect, perform operations analogous to the multiplication andsummation by weight vectors in discrete neural networks.

The continuous version of a projection neural network defined byEquation (16) can be extended by generalizing the notion of a radialbasis functions to neural manifolds. For discrete neural networks, afinite set of pattern vectors {x_(m)} and a radial basis function θ formthe nonlinear discriminate functions θ(∥x−x_(m)∥). The function θ hasits maximum value at the origin and the properties θ(x)>0 and θ(x)→0 as|x|→∞. Typically, θ is the Gaussian, exp(−x²/2σ²), or a similarfunction.

To transition from discrete basis functions to the continuous, we beginby replacing the discrete pattern vectors x_(m) with continuous densityfunctions ρ_(α). Each of the functions ρ_(α)(ξ,η,ζ) represents a“pattern” density defined on the input manifold. The patterns may bechosen from a finite set, or may represent samples taken from acontinuum of patterns. In general, this continuum could be anyparameterized set of functions over the spectral manifold J.

We may wish to associate a finite set of input patterns, ρ_(m), withparticular set of “target” or “label” functions q_(m) in the outputmanifold. Since the PEs have local support, a distributed collection ofPEs is required to cover the entire pattern density ρ_(m). Assume the PElattice is large enough so that many PEs are available for each patternand we assign a particular pattern to each PE which we label ρ_(i).

If we remove the convolution in time with the impulse response r(t) andomit the variables of integration (ξ,η,ζ) for h, and μ in (16) we have:

$\begin{matrix}{{q\left( {x,y,s} \right)} = {\sum\limits_{i}{\chi_{a}\left( {{\sigma \left( {\int_{H}{{\chi_{d}(h)}\ {\mu_{i}}}} \right)} \cdot {\tau_{i}\left( {x,y,s} \right)}} \right)}}} & (17)\end{matrix}$

The equation corresponding to a basis-function (q) neural network can beobtained by substituting either θ(χ_(d)(h) χ_(d)(ρ_(i))) or the simplerθ(h−ρ_(i)) for χ_(d)(h) in Equation (17), which results in:

$\begin{matrix}{{q\left( {x,y,s} \right)} = {\sum\limits_{i}{\chi_{a}\left( {{\sigma \left( {\int_{H}{{\theta \left( {h - p_{i}} \right)}\ {\mu_{i}}}} \right)} \cdot {\tau_{i}\left( {x,y,s} \right)}} \right)}}} & (18)\end{matrix}$

The processing elements PE(i) now have the additional property ρ_(i),which represents the pattern to which they are the most sensitive. Theintegral inside Equation (18) is maximum when h=ρ_(i) over the region ofintegration. This in turn maximizes the coefficient for the transmitterfunction τ_(i). The sum of a collection of transmitter functions {τ_(i)}associated with a particular input pattern ρ_(m) can then be defined toapproximate the desired “target” function q_(m), thereby creating therequired associations.

While many models that use action potentials as state variables formassociations using matrix operations on a large vector of neuronoutputs, equation (18) shows the neurotransmitter state model makes itpossible for a small number of neurons, even a single neuron, to recordan association between an input pattern ρ_(m)(ξ,η,ζ) and an outputpattern q_(m)(x,y,s).

The measures μ_(i) in Equation (18) can identify the regions where thepattern ρ_(m) is the most “sensitive”. For example, we can imaginephotographs of two different animals that appear very similar except fora few key features. The photographs, representing the two patterns ρ_(i)and ρ₂, would be approximately equal, but the various measures could betrained so that their value where the patterns were the same was small,but in the key regions where the patterns differed, they had much largervalues. In this way, even though the two image patterns are almost thesame, the output functions q_(m) that result from the integrals inEquation (18) could be very different.

The receptor measure and the transmitter function perform thecomplementary operations of converting back and forth between functionsdefined on a continuous manifold and discrete real values.

While the receptor measures {μ_(i)}, and transmitter functions {τ_(i)}are in general defined over local regions such as those shown in FIG. 8,it is worth emphasizing that they do not represent spectral functions orwavelets. They are however, defined over spaces where individual points(x₀,y₀,s₀) represent spectral functions on the isomorphic images. Theshape and sensitivity of the receptor measures do not correspond tospectral operators, but rather are the result of a process directedtoward learning arbitrary associations based on local information. Ingeneral, this process will yield asymmetric, one-of-a-kind, irregularfunctions.

A single layer neural network has limited computational utility andmultilayer networks are required to construct arbitrary associationsbetween functions. A two-layer discrete neural network and thecontinuous neural manifold, are shown in FIG. 9 a and FIG. 9 c.

Just as the integral is the continuous counterpart of the discretesummation, the neural manifold is the continuous counterpart of themultiple-layer neural network. The neural network illustrated in FIG. 9a forms associations between discrete vectors, while the neural manifoldillustrated in FIG. 9 c forms associations between continuous functions.At the cellular level, FIG. 9 b illustrates the processing elements(neurons) as discrete points in the function space N_(H,Q).

As we have seen, the measures {μ_(i)} in the neural manifolds canreplace the weights {ω_(i,j)} in the neural network. The same is alsotrue of the transmitter functions {τ_(i),}. In FIG. 9 b, each PE g_(i)(comparable to the g_(i) shown in FIG. 8) performs an operation similarto that defined by equations (17) or (18). The patterns {ρ_(i)} andmeasures {μ_(i)} analyze a region in the input manifold H and produce adiscrete value. This value can be compared to a node in the hidden layerof a discrete neural network. Since the transmitter functions can extendover a large area, even the entire output image, many differentprocessing elements may contribute to the value at any particular point(x,y,s). Consequently, the summations in equations (17) and (18) areequivalent to the summations in a neural network where the weightscorrespond to the values of the transmitter functions at any givenpoint.

The manifolds H and Q in FIGS. 9 b and 9 c represent the domains inwhich clouds of neurotransmitters would exist at some early point in theevolution of the nervous system. The processing elements, shown in FIG.9 b, represent neurons that perform an integration over the inputmanifold H using the receptor measure and disseminate the results in Qusing continuous overlapping transmitter functions. Separately, both thereceptor integration and the summation of transmitter functions performthe continuous analog of a single-layer neural network. Consequently,together they perform an operation analogous to a two-layer neuralnetwork.

The nodes of the neural network model are partitioned into the inputlayer, the hidden layer and the output layer. In the neural manifoldmodel, the input layer is analogous to the input manifold H and theoutput layer is analogous to the output manifold Q. Both H and Qrepresent the continuous distribution of neurotransmitters in physicalspace. The “hidden” layer is the space N_(H,Q), which equals theCartesian product of two function spaces, the space of all possiblemeasures on H and the space all possible output functions on Q. Theindividual neurons g_(i) are points in this infinite-dimensional productspace.

The interaction between the pre- and postsynaptic neuron across thesynaptic cleft is often modeled as a single real-valued weight w_(i,j)with the indices i and j enumerating the nodes that represent the preand postsynaptic neurons. At the cellular level illustrated in FIG. 9 b,nodes in the hidden layer N_(H,Q) still represent neurons, but nodes inthe input layer H, and the output layer Q, now represent the localconcentration (density) of neural transmitter at specific locations(x,y,s).

The collection of transmitter functions and receptor measures thatoperate within a single neurotransmitter cloud can also be viewed as atwo-layer neural network. In this formulation, the two-layer backpropagation algorithm now takes place between the pre- and postsynapticneurons with the errors propagating back from the receptor measures tothe transmitter functions.

In FIG. 9 b, the number of processing elements, g_(i), used in theconstruction of the NMs will affect the accuracy and capacity of the NM,but otherwise are abstractly separated from their intendedfunctionality, which is to associate functions representing continuousphenomena. This abstract association process is illustrated in FIG. 9 c.

Equations (17) and (18) express the computations of a neuron that issensitive to a single neurotransmitter. Given the variety of differentchemicals that act as neurotransmitters, both inhibitory and excitatory,we clearly need to extend the model to account for their effects. If wehave n different chemicals of interest in the extracellular space we canmodel their concentration at each point as vector with the direct sumh(x,y,s)=h₁(x,y,s)⊕h₂(x,y,s) ⊕ . . . ⊕h_(n)(x,y,s). Under the assumptionthat all of the all these substances act independently and their neteffect on depolarizing the cell membrane is additive, we can extend thereceptor measures the patterns ρ_(i) and the transmitter functions τ_(i)in the same way and perform the integration and summation operationswithin (17) and (18) separately. Nonlinear interactions between theneurotransmitters on the dendritic tree cell membrane will require theappropriate modifications to the integral over the manifold H.

When samples of a continuous function that is defined on a space of highdimension are arranged in a space of much lower dimension, the sampleswill in general appear to be discontinuous. Consequently, when acollection of neurons {g_(i)} representing samples taken from theinfinite-dimensional space N_(H,Q) are arranged in three-dimensionalphysical space, the result will look as if it is discontinuous. Theresulting firing rates may appear to be stochastic when in fact they aredeterministic. Moreover, realistic neural-field models that attempt todescribe the observed firing rates of large groups of neurons as acontinuous function in physical space will be difficult or impossible tocreate.

If M is an n-dimensional manifold, for a function ƒεL²(M) and a scalevector sεR₊ ^(m), the spectral manifold over which S[f](x,s) is definedhas dimension n+m. Consequently, the transformed function isover-specified. Because of the redundancy, it is not true that anyfunction hεL²(M×R₊ ^(m)) is the spectral transform of some functionƒεL²(M).

To gain an intuitive understanding why this is the case, imagine the x−yplot of the one-dimensional function ƒ(x) that would be drawn in atrigonometry classroom. The resulting image has black pixels near thepoints (x,f(x)) and white pixels elsewhere. While it may be true thatall one-dimensional plots can be drawn as two-dimensional images, it isnot true that any image—for example a photograph taken outdoors—can berepresented as a one-dimensional plot. While it is possible to constructcontinuous, dense, space-filling curves having infinite length (Munkres,2000), in general, an arbitrary two-dimensional photograph does not havean “inverse” that corresponds to the plot of a one-dimensional analyticfunction.

The PEs in FIG. 4 compute the required outputs for a large number ofstored associations based on only limited local information.Consequently, the overall result of these calculations can only be anapproximation, which may not have a well-defined inverse transformation.However, using the reproducing kernel it is possible to estimate theapproximation error and calculate the closest function for which theinverse spectral transformation exists.

For the one-dimensional case (n=m=1), the following equation defines thenecessary and sufficient conditions for a function to be a wavelettransform (Mallat, 1999).

$\begin{matrix}{{{\lbrack f\rbrack}\left( {x,s} \right)} = {\frac{1}{C_{\phi}}{\int_{0}^{+ \infty}{\int_{- \infty}^{+ \infty}{{\lbrack f\rbrack}\left( {\xi,\eta} \right){K\left( {x,s,\xi,\eta} \right)}\ {\xi}\frac{\eta}{\eta^{2}}}}}}} & (19)\end{matrix}$

where the constant C_(φ) is given by Equation (4). The reproducingkernel K measures the correlation between the wavelets φ_(x,s)(α) andφ_(ξ,η)(α) and is defined by

$\begin{matrix}\begin{matrix}{{K\left( {x,s,\xi,\eta} \right)} = {\int_{- \infty}^{+ \infty}{{\phi_{x,s}(\alpha)}{\phi_{\xi,\eta}(\alpha)}\ {\alpha}}}} \\{= {\langle{\phi_{x,s},\phi_{\xi,\eta}}\rangle}}\end{matrix} & (20)\end{matrix}$

Let E=L²(

×

₊) and let U denote the linear subspace of E where the inverse wavelettransform exists. Using the reproducing kernel specified by (20) wedefine the linear operator V by

$\begin{matrix}{{{\lbrack f\rbrack}\left( {x,s} \right)} = {\frac{1}{C_{\phi}}{\int_{0}^{+ \infty}{\int_{- \infty}^{+ \infty}{{f\left( {\xi,\eta} \right)}{K\left( {x,s,\xi,\eta} \right)}\ {\xi}\ \frac{\eta}{\eta^{2}}}}}}} & (21)\end{matrix}$

From (19), we note that for fεU, V[f]=f. In a straight forward proofthat will not be repeated here, it can be shown that V is an orthogonalprojection of E onto U. If we view the local estimation errors in thecalculations as additive noise w(x,s), then

V[f+w]=f+V[w]  (22)

Since V is an orthogonal projection, ∥V [w]∥≦∥w∥. That is, V removes thecomponent of the noise that is in U^(⊥) and thereby projects theestimate to a function that is closer to the correct solution f.

From the definition of the reproducing kernel (20), we see that at afixed position (x₀,s₀) in the spectral manifold, the kernel K(x₀,s₀,ξ,η)is zero or near zero for values of (ξ,η) where the spectral functionsφ_(x) ₀ _(,s) ₀ and φ_(ξ,η) do not overlap. Moreover, K is defined interms of the wavelets themselves and does not depend on the transformedfunction ƒ. Consequently, at any point in the spectral manifold, we canpre-compute the function K, which goes to zero outside of a localneighborhood of (x₀,s₀). Note that for low-frequency spectral functions,the overlapping regions will be large, but if the spectral functions arenormalized, their inner product will still go to zero as the distancebetween (x₀,s₀) and (ξ,η) increases.

In multilayer-projection-neural networks, the computation in each layerequals a vector multiplied by a matrix followed by a nonlinear transferfunction. If the nonlinear-transfer function is replaced by a linearone, then the matrices can be pre-multiplied and the multilayer networkbecomes a single layer network. An analogous result occurs if one of thetransfer functions, σ, χ_(d) or χ_(a), are replaced with a linearoperation in a multilayer neural manifold calculation. This allows theintegral and summation signs to be interchanged and various interactionsto be pre-computed. Although this may reduce the discriminating capacityof a multilayer network, it can also allow the effect of the reproducingkernel on the transmitter functions to be calculated in advance. For aone-dimensional function, if we replace the function χ_(a) withmultiplication by a constant c in (17) we have

$\begin{matrix}{{q\left( {x,s} \right)} = {\sum\limits_{i}{c \cdot {\sigma \left( {\int_{H}{{\chi_{d}(h)}\ {\mu_{i}}}} \right)} \cdot {\tau_{i}\left( {x,s} \right)}}}} & (23)\end{matrix}$

If we now use the reproducing kernel K to project this onto the solutionspace of possible wavelet transforms we have

$\begin{matrix}\begin{matrix}{q = {{\lbrack q\rbrack}\left( {x,s} \right)}} \\{= {\frac{1}{C_{\phi}}{\int_{0}^{+ \infty}{\int_{- \infty}^{+ \infty}\left( {\sum\limits_{i}{c \cdot {\sigma \left( {\int_{H}{{\chi_{d}(h)}\ {\mu_{i}}}} \right)} \cdot {\tau_{i}\left( {\xi,\eta} \right)}}} \right)}}}} \\{{K\left( {x,s,\xi,\eta} \right)\ {\xi}\ \frac{\eta}{\eta^{2}}}} \\{= {\sum\limits_{i}{{\sigma \left( {\int_{H}{{\chi_{d}(h)}\ {\mu_{i}}}} \right)} \cdot {\begin{pmatrix}{\frac{c}{C_{\phi}}{\int_{0}^{+ \infty}{\int_{- \infty}^{+ \infty}{\tau_{i}\left( {\xi,\eta} \right)}}}} \\{K\left( {x,s,\xi,\eta} \right)\ {\xi}\ \frac{\eta}{\eta^{2}}}\end{pmatrix}.}}}}\end{matrix} & (24)\end{matrix}$

That is, we can pre-compute the integral transform of the reproducingkernel on each of the transmitter functions τ_(i), and then sum over theresults.

So far, we have discussed the reproducing kernel only for case ofone-dimensional wavelets φ_(x,s). For the multidimensional case, we notethat the definition (20) is expressed as an inner product on R which canbe easily extended to the inner product on any spectral manifold J. Ingeneral, reproducing kernels require only the mathematical structure ofa Hilbert space.

Linear transforms have properties analogous to those defined by S above.These transforms include frame operators and the dyadic wavelettransform. The discrete windowed Fourier transform and the discretewavelet transform are both examples of the mathematics defined by frametheory.

We use the term “linearly transformed space” to refer to the space ofinner products {

f,φ_(γ)

}_(γεΓ) which characterize any signal f defined on a Hilbert space H.Note that H may be discrete or continuous and the set F may be infinite.

Frame theory (Mallat, 1999) describes the completeness and redundancy oflinear signal representations. A frame is a collection of vectors{φ_(n)}_(εΓ), where Γ is a subset of the integers, that characterizes asignal f from its inner products {

f,φ_(n)

}_(nεΓ). We define b[n]=

f,φ_(n)

. The sequence {φ_(n)}_(nεΓ) is a frame on H if there exist twoconstants A>0 and B>0 such that for any fεH

$\begin{matrix}{{A{f}} \leq {\sum\limits_{n \in \Gamma}{{\langle{f,\phi_{n}}\rangle}}^{2}} \leq {B{f}}} & (25)\end{matrix}$

For any frame, we define the “frame operator” T as T[f](n)=b[n]=

f,φ_(n)

. If the constant A is strictly greater than one, then therepresentation defined by the frame operator always contains redundantinformation.

If T is a frame operator, we denote by T* the adjoint of T defined by

Tf,x

=

f,T*x

where x is a sequence of real numbers in l²(Γ). For any frame{φ_(n)}_(nεΓ) the dual frame is defined by

{tilde over (φ)}_(n)=(T*T)⁻¹φ_(n)  (26)

and a pseudo inverse for T can be defined by

{tilde over (T)} ⁻¹=(T*T)⁻¹ T*  (27)

The dual frame allows us to define the reproducing kernel as

K=

{tilde over (φ)}_(p),φ_(n)

  (28)

A vector b[n] is a sequence of frame coefficients if and only if

$\begin{matrix}{{b\lbrack n\rbrack} = {\sum\limits_{p \in \Gamma}{{b\lbrack p\rbrack}{\langle{{\overset{\sim}{\phi}}_{p},\phi_{n}}\rangle}}}} & (29)\end{matrix}$

For any b[n] which is not a sequence of frame coefficients, thisequation defines an orthogonal projection V onto the nearest validsequence of coefficients.

The association operations in a PsyMAP may take place in a linearlytransformed space that is not a frame. However, frames guarantee thatthe inverse transform exists and that it is possible to define anorthogonal projection based on the reproducing kernel.

Neural network models usually form a matrix where the input neuronsindex the columns, the output neurons index the rows, and theirinteraction weights are the matrix coefficients. From an engineeringpoint of view, a simple straightforward implementation of this matrixformulation for neural manifold will be problematic since for evenmoderate resolution one megapixel images, the matrix will have 10¹²elements. While this matrix will be sparse due to the local connectivityof the neurons, because of the low-frequency spectral functions,multiplication and inverse operations will quickly result in non-zerovalues throughout the matrix. In fact, part of the importance oflow-frequency wavelets in the recursive Ψ-MAP design, is that they helpspread calculations performed in one small area over the entire image.

At the various levels of abstraction, it is important to preserve theconnection to the underlying differential structure of the imagemanifolds. This mathematical structure is essential in maintaining thecoherence with real-world natural phenomena and in providing thefoundation for calculations such as the orthogonal projection based onthe reproducing kernel. Digital simulations and general-purposeimplementation require that that the manifolds be discretized, but thereare many trade-offs to be made in determining the best resolution. Evenwithin a single application, there may be advantages to maintainingrepresentations at multiple resolutions. For example, we may wish toperform one operation such as learning at a high-resolution, while, forefficiency, the association formation (classification) process takesplace at a lower resolution.

Although asynchronous design techniques have been around for some time,almost all digital computers utilize synchronous architectures thatemploy a clock signal to coordinate operations. Without a clock signal,a complex arrangement such as the Ψ-MAP Array shown in FIG. 6 would bedifficult to control and prone to errors. Slightly unequal processingtimes for individual neurons or groups of neurons would inevitablyoccur, as would variations in transmission times due to differences inphysical proximity. Over time, these errors would tend to be cumulative.An image, or the spectral representation of an image, only has meaningwhen all of the pixels or voxels maintain temporal coherence.Consequently, the necessity of processing and transmitting entirespectral manifolds suggests the need for a system clock, presumablyarising from a central location such as the thalamus.

In the digital domain, the system clock is a regular pulse train thatquickly transitions between the states zero and one. In the analogdomain, however, the system clock can contain phase information that mayfacilitate computations on manifolds. For example, suppose the systemclock is a periodic triangular wave, which rises linearly from itsminimal value to its maximal value and then falls linearly to repeat thepattern. For reference, we will refer these two “sides” of thetriangular wave as the rising and falling intervals.

In the definition of the spectral manifolds, we mapped the scalingparameter sε(0,∞) monotonically to the interval (0,z₀) that correspondsto the physical thickness of a single cortical layer. We now extend thiscorrespondence by continuously mapping the interval (0,z₀) to theinterval between the minimal and maximal values of an idealizedtriangular wave system clock. In effect, the clock now sweeps back andforth through the spectral manifold thickness from the low-frequencyspectral functions to the high-frequency spectral functions. Atriangular wave clock signal can coordinate the implementation ofprogressive resolution algorithms that begin with an approximatesolution based on the low-frequency information and progressively addthe details associated with the high-frequency spectral functions.Alternately, following Marr's principle of least commitment (1976,1983), we could make small incremental decisions starting with the localhigh-frequency details and progress to larger areas covered by thelow-frequency wavelet functions.

A single-layer general association NM G can be recast as a progressiveresolution algorithm in the following way. Label the correspondingscaling factor at clock phase θ as s_(θ). Modify G so that as the clocksweeps through the spectral manifold, at a particular clock phase θ, Gonly accepts the spectral inputs whose scaling parameters are in theinterval I=(s_(θ),∞) [alternately I=(0,s_(θ))]. When the system clockreaches its extremum, I=(0,∞) and the output of a single-layerprogressive resolution algorithm would be the same as a non-progressiveversion. Although a single-layer, progressive refinement NM consideredin isolation would generate the same results as before, when used in arecursive design such as a Ψ-MAP, during the association process it cangenerate entirely different results.

In the previous discussion, we did not explicitly assign the minimum andmaximum values of the system clock to the low or high frequencyfunctions, and it may be that different classes of neurons make oppositeassignments. Thus, during the rising and falling intervals of ouridealized triangular wave clock, two separate computations may beperformed, both of which use a progressive resolution method that startswith the high frequencies and progresses to the low frequencies. Thisallows us to design systems using a two-phased triangular clock as thecontrol signal, but still maintain flexibility in the choice ofalgorithms.

Non-linear approximation methods can provide insight into encodingschemes that can represent arbitrary images with a relatively smallnumber of predefined spectral components. These non-linear approximationalgorithms generally assume an orthonormal basis for the wavelets. Wedescribe how to extend these methods to find approximations in thegeneral case of continuous spectral transforms. However, there is noguarantee that the inventive algorithms will produce optimal results, asis true for some methods that employ a wavelet basis.

Let q be a function defined on the spectral manifold Q and use theorthogonal projection V to calculate q, the function closest to q forwhich an image transform exists.

q=V[ q]  (30)

Denote the set of available pyramidal neurons in Q as Γ={γ}, where eachneuron is identified with the spectral function φ_(γ) having parameters(x_(γ),y_(γ),s_(γ)). We will assume that the spectral functions {φ_(γ)}are normalized. Let q_(γ)=q(x_(γ),y_(γ),s_(γ)) be the value of qcorresponding to each pyramidal neuron.

Let f_(q) be the image that corresponds to q. That is

f_(q)=S⁻¹[q]  (31)

We wish to find a small finite subset Ω, Ω⊂Γ that approximates f_(q)with an error bound of ε. The approximation f_(Ω) corresponding to theset of neurons Ω is given by

$\begin{matrix}{f_{\Omega} = {\sum\limits_{\alpha \in \Omega}{q_{\alpha}{\phi_{\alpha}.}}}} & (32)\end{matrix}$

We wish to minimize the error

$\begin{matrix}{\sup\limits_{\gamma \in \Gamma}{{\langle{{f_{q} - f_{\Omega}},\phi_{\gamma}}\rangle}}} & (33)\end{matrix}$

A simple, straightforward algorithm is to start with Ω equal to the nullset, and to iteratively perform the following two steps:

1. Select the neuron βεΓ which has the maximal (or near maximal) valueof

|

f_(q)−f_(Ω),φ_(β)

|  (34)

and add β to the set Ω.

2. If the error is below the threshold

${\sup\limits_{\gamma \in \Gamma}{{\langle{{f_{q} - f_{\Omega}},\phi_{\gamma}}\rangle}}} < ɛ$

or the maximum number of iterations has been exceeded then quit,

otherwise repeat.

In step two, depending on q and the {φ_(γ)}, the error value may notconverge to zero. Therefore, we include test for a maximum number ofiterations. In step one, it may appear that on each iteration, we mustrecompute all of the inner products of f_(q)−f_(Ω) with all of the{φ_(γ)}, however

$\begin{matrix}\begin{matrix}{{\langle{{f_{q} - f_{\Omega}},\phi_{\gamma}}\rangle} = {\langle{{f_{q} - {\sum\limits_{\alpha \in \Omega}{q_{\alpha}\phi_{\alpha}}}},\phi_{\gamma}}\rangle}} \\{= {{\langle{f_{q},\phi_{\gamma}}\rangle} - {\sum\limits_{\alpha \in \Omega}{q_{\alpha}{\langle{\phi_{\alpha},\phi_{\gamma}}\rangle}}}}} \\{= {q_{\gamma} - {\sum\limits_{\alpha \in \Omega}{q_{\alpha}{\langle{\phi_{\alpha},\phi_{\gamma}}\rangle}}}}}\end{matrix} & (35)\end{matrix}$

Thus, when a neuron β is added to the set Ω, all that is necessary is tosubtract the values q_(β)

φ_(β),φ_(γ)

from the surrounding pyramidal neurons. However, the value

φ_(β),φ_(γ)

is precisely the value of the reproducing kernel, K, and can bepre-computed. That is, the values can be hard-wired in the neuronalinterconnections with preset weights. Presumably, this subtraction isanalogous to the lateral inhibition between pyramidal neurons that isknown to take place when a pyramidal cell fires.

It is possible to design several different types of digital flip-flopsusing simple binary logic gates. It is also possible to design severaltypes of Ψ-MAPs using neural manifolds. FIG. 10 shows a detailed Ψ-MAPdesign that illustrates one possible architecture. Variations of thisdesign can achieve the same or similar functionality, and it is clearthat additional research will be required to determine the one used inthe neocortex. Nevertheless, hypothesizing one or more concrete designsallows us to move forward by simulating, analyzing, and modifyingcandidate architectures. Some of the alternative architectures willexhibit slightly different input and output characteristics and byfocusing on a small number of Brodmann areas, carefully formulatedpsychological experiments may be able to discriminate between some ofthe possible designs.

The detailed Ψ-MAP computational model in FIG. 10 illustrates thegeneral pattern of interconnection. G denotes an association operation,M denotes the masking and multiplexing operation and V denotes theorthogonal projection based on the reproducing kernel. Each of the sixspectral manifolds, C, H, Q, D, L and P are layered one-on-top the otherto form the lamina of a single Brodmann area in the neocortex.Consequently, the connections between them are local and form the basisof the cortical columns.

The double lines in FIG. 10 represent the transfer of data defined onspectral manifolds. The transformations marked G, V and M representdifferent types of neural manifolds. The letter G denotes a generalassociation NM, the letter V denotes orthogonal projections based on thereproducing kernel and the vertical box marked M performs multiplexingoperations based upon the masks derived from the thalamic inputs markedT_(S) and T_(R). Note that each NM labeled G is trained with a differentset of input and output relationships and consequently carries a uniquesubscript that identifies it as a unique transform. However, the NMslabeled V and M, do not carry subscripts since they perform a fixedoperation that is the same in all instances.

All computations require some amount of time Δt to finish, and even whenit is not explicitly included in the equations, it is assumed that theNMs vary with time. We could make this time dependency explicit byincluding it in all of the equations, for example, q(x, y, s,t÷Δt)=G_(H)[h(x,y,s,t)]. However, to simplify the discussion, we removethe time-dependent behavior by assuming that the Ψ-MAP is stable andthat the inputs have remained constant sufficiently long for the Ψ-MAPto reach a steady state.

The inputs, {s_(k)} and {r_(k)}, represent collections of spectralfunctions that arise from either subcortical regions or from the outputsof other Brodmann areas. The integration of data from three separateinput images to form a single output pixel was illustrated in FIG. 4.However, in the context of the neuron model presented in FIG. 8, we seethat the receptor measure μ is actually three separate measures {μ_(k)},one for each image. Mathematically we can view the collection of imagesas a single vector-valued image with a single measure μ=μ₁⊕μ₂⊕ . . .⊕μ_(k).

Since μ represents the sensitivity of a particular neuron's dendritictree to a continuous “cloud” of neurotransmitter, if the transmitterfunctions from k separate images were released into the open physicalspace of the same manifold, the clouds of neurotransmitter would overlapand it would be impossible for the neuron to match the receptor measureμ_(k) with the appropriate input image s_(k) or r_(k). While it ispossible for a dendritic tree to have varying sensitivity to severaldifferent neurotransmitters that represent the components of avector-valued function, this conjecture may be unnecessary. Instead,synapses could establish an effective mechanism for separating thecomponent images.

The synaptic cleft localizes the action of the neurotransmitter to thesmall region between the pre and postsynaptic neurons. A single synapseformed during the learning process establishes a connection to onespecific input image. From the point of view of the postsynaptic neuron,once the connections have been established, there is no way todistinguish which synapses are associated with which inputs. However,from an overall systems point of view, the dendritic computation couldbehave like separate integrations over the vector space formed by thedirect sum of image components and receptor measures.

If G_(S) (or G_(R)) forms the same output association for severalmembers of a given collection of inputs {s_(k)}; s_(k)εL²(S_(k)) (or{r_(k)}; r_(k)εL²(Rk)), then the associations will mutually reinforceone another since for any receptor measure μ

$\begin{matrix}{{{\int_{S}{s\ {\mu}}} = {\sum\limits_{k}{\int_{S}{s_{k}{\mu_{k}}}}}}{and}} & (36) \\{{\int_{S}{r\ {\mu}}} = {\sum\limits_{k}{\int_{S}{r_{k}{\mu_{k}}}}}} & (37)\end{matrix}$

Consequently, even though single inputs may not be sufficiently strongto bring forth a recollection, multiple inputs will add “context”, andtheir combined effect may surpass a threshold required to evoke theentire memory.

The overall pattern of interconnection for the Ψ-MAP shown in FIG. 10 isthe same as that shown in FIG. 3 but we have now filled in the internaldetails of the Λ-MAPs. The Ψ-MAP shown contains eight general manifoldassociation processors labeled G, divided evenly between the externaland internal Λ-MAPs, Λ_(E) and Λ_(I). Each of these can be considered tobe analogous to a single-layer neural network, that is a summationfollowed by a nonlinear transfer function. This single-layer computationmay be performed in either the dendritic tree or the axonal tree.

We use the product notation to denote the composition of transforms, andwrite G_(H)G_(P)[p] for G_(H)[G_(P)[p]], which is the output of firsttransforming p by G_(P) and then transforming the result by G_(H). InFIG. 10 we assume that the composition of these two transformscorresponds to the synapses between the axonal trees and dendritic treeswithin one neurotransmitter layer, but it could also correspond to thecalculations in the dendrites and axons of a single layer of neurons. Wewill use the term neural manifold to denote the operation analogous toboth single and multi-layer neural networks.

In Λ_(E), the NM G_(C)G_(S) transforms the inputs {s_(k)}, whileG_(H)G_(P) transforms the recursive input from P. Symmetrically, inΛ_(I), the NM G_(D)G_(R) transforms the inputs {r_(k)}, while G_(L)G_(Q)transforms the recursive input from Q.

The multiplexor masks allow the thalamus to focus attention on selectedregions and to control whether individual Brodmann areas accept newinputs or retain their current contents. Each of the reciprocal Λ-MAPs,shown in FIG. 10, contains two separate, NMs whose outputs feed into amultiplexor box labeled M. A third input, shown with a diagonal line, isthe control mask, which is marked T_(S) in the exterior Λ-MAP and T_(R)in the interior Λ-MAP. Both multiplexors perform the same operation, andinside the box the two association inputs are labeled A and B. Forreference however, we will label the Λ_(E) inputs b_(P) and a_(S), andthe Λ_(I) inputs b_(Q), and a_(R). From FIG. 10 we see that

a_(S)=G^(C)G_(S)[{s_(i)}]

a_(R)=G^(D)G_(R)[{r_(i)}]  (38)

and

b_(P)=G^(H)G_(P)[{P}]

b_(Q)=G^(L)G_(Q)[{q}]  (39)

To distinguish between the two separate general association neuralmanifolds inside each Λ-MAP that process the two separate input images,we will refer to a NM that corresponds to either a_(S) or a_(R) in FIG.10 as an input NM and a NM that corresponds to either b_(P) or b_(Q), asa latch NM.

Let α(x,y,s) be one of the thalamic mask signals T_(S) or T_(R). Thesecan control the multiplexor by performing an operation analogous to theimage composition alpha blending calculation used in computer graphics(Foley, van Dam, Feiner and Hughes, 1990; Thompson 1990). For αε[0,1]the result of the alpha blending operation is given by

output=(1−α)A+αB  (40)

When α=0, the output equals A, and when α=1, the output equals B. Valuesin between smoothly blend the two images.

If we want the ability to force the output to zero, during the twophases of the clock we can transmit a second mask β(x,y,s) which wesimply multiply by the output. Thus, when β is equal to zero everywhere,the output will be forced to zero as well.

At this point, in what we might call “the masking problem”, we find atradeoff between computation and accuracy. The problem arises because wewould like to perform a bitwise mask of the image data, but the datathat is available is the spectral representation of the images. UnlikeFourier transforms, the multiplication of two wavelet transforms doesnot correspond to a convolution operation on the untransformed data(Pérez-Rendón & Robles, 2004). If we want to perform a bitwise maskingoperation, then during each time step, for a spectral operator S, wewould have to compute the inverse and forward calculations:

M[A,B,α,β]=S(β((1−α)S ⁻¹ [A]+αS ⁻¹ [B]))  (41)

An alternative is to proceed with the multiplication on the spectraldata and accept the resulting indistinct “blurriness” that results. Athigher frequencies, the masks will be sharper since these spectralfunctions have smaller support. The complete expression for thepointwise masking operation is then:

M[A,B,α,β](x,y,s)=β(x,y,s)((1−α(x,y,s))A(x,y,s)+α(x,y,s)B(x,y,s)  (42)

Since the spectral manifold has a higher dimension (three) than theoriginal image space (two), there are many spectral functions, for whichthere is no corresponding image. Mathematically, almost all functions hdefined on the spectral manifold H, do not have an inverse S⁻¹[h].

The result of an association formed in a two layer real-valued neuralnetwork will undoubtedly contain small errors that result in “invalid”spectral functions. Moreover, in general, a masking operation such asthe one defined by Equation (42), will also result in a spectralfunction that does not correspond to the transform of an actual image.

Images used as masks do not need to be generated by a separate controlmechanism and can in fact be generated by other Ψ-MAPs in the Ψ-MAParray. The capability to generate masks within the array itselfsignificantly extends its power and flexibility.

Fortunately, using the reproducing kernel defined in Equation (20) wecan project these functions using the linear operator V given inEquation (21) to the nearest function for which the inverse transformS⁻¹ produces a valid result. This operation is shown in FIG. 10following the multiplexing operation. Since the results of theorthogonal projections are Q and P, we thereby guarantee that theoutputs of a Ψ-MAP always correspond to actual images.

As long as the control signals α(x,y,s) and β(x,y,s) in Equation (42)are identically equal to one, in both the exterior and interior Λ-MAPs,the Ψ-MAP will ignore its inputs and retain its current stored value.Under these conditions, the Ψ-MAP is latched and the contents of thisshort-term memory remain fixed on the outputs q and p. During this time,the recursion equations that regulate the operation of the Ψ-MAP are:

q=VG_(H)G_(P)[p]

p=VG_(L)G_(Q)[q]  (43)

Even though FIG. 10 does not show any clock signals, similar to theground signal in a conventional circuit diagram, we assume that a clockis available to any component that requires it. Suppose an analog, (e.g.triangular wave) clock is used by the NMs G_(Q), G_(P), G_(H), G_(L),G_(S), and G_(R) to implement a progressive resolution algorithm. Duringa single clock cycle, the computation may start with associations formedfrom the low frequency (or high frequency) components of the image andgradually proceed by adding spectral information in the progressivelyhigher (or lower) frequency ranges.

During classification, when the Ψ-MAP is accepting inputs and formingnew associations, the use of progressive resolution algorithms may haveadvantages and disadvantages. If we consider f=(q,p,h,l,c,d) as a pointin the product space of functions defined on Q, P, H, L, C, D then ffollows a trajectory from an initial point determined by the inputs andprevious state to a new “solution” state. An advantage of theprogressive resolution method is that this trajectory to a fixed pointbecomes more predictable. A possible disadvantage is that the trajectorymay predictably lead to the wrong solution. This may result when two ormore reciprocal image pairs have inputs that appear similar at low orhigh frequencies but are associated with completely different outputs.Thus, the input NMs may need to accept a range of frequencies that issufficient to cause the paths to bifurcate, when needed, during theearly stages of the classification process.

When a Ψ-MAP is latched, in a steady-state condition, if we combine thetwo Equations in (43) we have:

q₀=VG_(H)G_(P)VG_(L)G_(Q)[q₀]  (44)

That is q₀ will be a fixed point of this six-layer NM transformation.

We define

Z≡VG_(H)G_(P)VG_(L)G_(Q)

and label successive outputs formed as part of the association processas q_(i). Thus

q_(i+)=Z[q_(i)]  (45)

If for c<1

∥q _(i+) −q _(i) ∥=∥Z[q _(i) ]−q _(i) ∥≦c∥q _(i) −q _(i−1)∥  (46)

in a complete subspace U, then Z is a contraction and by the contractionlemma (Spivak, 1979) we are guaranteed that there exists a unique fixedpoint, q₀, in U. Note that since the metric space must be complete, thatis all Cauchy sequences must converge, in some cases, it may benecessary to form the closure the original space (Kolmogorov & Fomin,1970; Royden, 1988).

The existence of regions where Z is a contraction are important becausethey imply stability. If Z is a contraction in a subspace U thatcontains an open neighborhood about q₀, then even if there is a smallamount of added noise, the Ψ-MAP will follow a trajectory back to q₀.

Learning at the level of synapses takes place by modifying sensitivityof the dendritic tree and by modifying the release of transmitters bythe axonal tree. The learning algorithms used for training neuralnetworks have similar counterparts in neural manifolds. Several neuralnetwork methods use a gradient decent algorithm based on asum-of-squares error function E. Suppose an NM G, composed of PEs{g_(i)}, must associate a finite set of input functions {h_(β)}⊂L²(H)with the output functions {q_(β)}⊂L²(Q). Based on Equation (18), understeady-state conditions, we can express the total error as

$\begin{matrix}\begin{matrix}{{{E()} = {\sum\limits_{\beta}{\int_{Q}\left( {{\left\lbrack h_{\beta} \right\rbrack} - q_{\beta}} \right)^{2}}}}\ } \\{= {\sum\limits_{\beta}{\int_{Q}{\begin{pmatrix}{{\sum\limits_{i}{\chi_{a}\begin{pmatrix}{\sigma {\left( {\int_{H}{{\theta \left( {h_{\beta} - p_{i}} \right)}\ {\mu_{i}}}} \right) \cdot}} \\{\tau_{i}\left( {x,y,s} \right)}\end{pmatrix}}} -} \\{q_{\beta}\left( {x,y,s} \right)}\end{pmatrix}^{2}\ {x}{y}{s}}}}}\end{matrix} & (47)\end{matrix}$

To evaluate these equations, we convert the integrals to sums bydiscretizing the spectral manifolds H and Q. If the receptor measuresand transmitter functions are non-zero on only small local regions in Hand Q respectively, the computation required to estimate the integralsis significantly reduced.

When the input manifold H is converted to voxels, the receptor measuresand transmitter functions are both converted to a set of weights{w_(α)}. Once discretized, we can use standard neural network algorithmsto search for an error minimum where

$\begin{matrix}{\frac{\partial E}{\partial w_{\alpha}} = 0} & (48)\end{matrix}$

In converting the integral to a sum, we restricted the region ofintegration to locations where the receptor measure was non-zero.Although the resulting weights will be non-zero only in this smallregion, we may wish to expand the region slightly to include voxels(adjacent to the dendritic tree or the axonal tree) where the weightshave a zero value. Doing so may increase the accuracy and capacity ofthe Ψ-MAP, at the cost of increasing the computation required todetermine associations. If the reception measure is a model of aphysical neuron, the weights will be non-zero only on or near thesurface of the dendrites or axons. By including the voxels that areadjacent to the surface, but have a zero weight, we can simulate thephysical three-dimensional growth of a neuron as it learns newassociations.

Since a conventional two-layer neural network has a single input vectorand a single output vector, training the network is simply a matter offorming associations between one set of inputs and one set of outputs. AΨ-MAP on the other hand has two sets of input images, two sets of outputimages, two control masks and four neural manifolds. We will discuss sixdifferent types or categories of learning. For each category, wedescribe only one example with the understanding that within eachcategory other similar cases are possible, the most obvious oncecorresponding to reversing the roles of the exterior and interiorΛ-MAPs. Each of the six categories describes the training of a singleNM. The CNS presumably combines several of these “elemental” learningmethods to form complex “composite” learning mechanisms, both within asingle Ψ-MAP, as well as between multiple processors in a Ψ-MAP array.

In FIG. 11, we have shown the previously discussed connections with thinlines and added heavy double lines for the image connections used inlearning. Also shown are feedback paths for q and p from the output ofthe projection operator V. Including the control mask, the multiplexingbox M has three separate inputs, A, B and T. The difference between anytwo of these inputs, A−T, B−T, A−B or A−(p or q) can act as an errorterm that feeds back from M to train the general association NMs. Forsimplicity, the Fig. shows a single pathway, which transmits any one ofthe various difference terms back from the multiplexor to the NMs;however, it is clearly possible to have separate pathways for theseparate difference terms. Analogous to a two-layer neural network, theerror from the second half of a NM must propagate back to the firsthalf.

FIG. 11 is a diagram illustrating the Ψ-MAP learning pathways, that areillustrated with heavy double lines, and feed the three-dimensionalerror signals used for training back to the input and latch NMs of theexterior and interior Λ-MAPs. The line feeding back from the multiplexorM can carry any of the x−T, A−B or x−q (x−p) difference signals, where xrepresents either the A or B multiplexor inputs. The difference betweenthe input and the output of the orthogonal projection operator is theReproducing Kernel Error (RKE) and measures the amount by which themultiplexor output differs from the nearest possible image.

In the first of the six types of learning, we train an input NM based onan image from the thalamus. During the normal operation previouslyillustrated in FIG. 10, the inputs T_(S) and T_(R) were control signals.Now however, they function as data signals. For some arbitrary inputs_(k), we can train the exterior Λ-MAPs to generate the output q₀=V[T_(S)] simply by feeding back the error signal a_(S)-T_(S), and usingit to train the Λ_(E) input NMs. A symmetric operation in the reciprocalinterior Λ-MAP can associate ρ₀=V [T_(R)] with some input r_(k).

Suppose the input NMs of Λ_(E) associates an input s_(k) with an outputq₀, and the latch NMs of Λ_(I) associate an input q₀ with some outputρ₀, that is, q₀=VG_(C)G_(S)[s_(k)] and p_(G)=VG_(L)G_(G)[q₀]. If we wantto retain the output pair (q₀, p₀) in short-term memory when the inputs_(k) is no longer present, we must train the latch NMs of Λ_(E) toassociate the input p₀ with the output q₀. While s_(k) is still present,the multiplexor error signal A−B contains the necessary information totrain the Λ_(E) latch NMs so that q₀=VG_(H)G_(P)[p₀].

Note that if the Λ-MAPs have two separate feedback paths for the A−T andB−T error terms, and we present inputs s₀, r₀, T_(S), and T_(R) to theΨ-MAP at the same time, it is possible to train all four NMssimultaneously. In this case, either or both of the inputs s₀ or r₀, canbe used to recall the associated reciprocal image pair V [T_(S)], 92[T_(R)].

In the third category of learning, assume that the input NMs of Λ_(E)associates an input s_(k) with an output q₀, and both Λ-MAP latch NMsare trained to retain the output pair (q₀, p₀) when s_(k) is removed. Asecondary association is a relation between this same output pair (q₀,ρ₀)—whatever it may be—and other inputs. If the Ψ-MAP is latched, it canlearn secondary associations by using the error signals A−B to train theinput NMs when presented with one or more other inputs {r_(i)} or{s_(i)}.

Suppose rather than data images, we again use the thalamic inputs toperform an image masking operation. Following the masking operation andthe projection V, we can feed the output of V, representing a maskedimage subregion, back to the input and latch NMs as the error signal x−qor x−p, where x is either of the multiplexor inputs A or B. By using thesame learning methods as before, we can now train the NMs to associate aparticular subregion of an image with some input image or the reciprocalΛ-MAP output.

As is the case with a neural network, in a general association NM thereis no direct path from the input to the output. We refer to a pathwaythat bypasses one of the four general association NMs as a shunt.Without shunts, it is not possible to copy one of the inputs {r_(i)} or{s_(i)} directly to the outputs p or q. In FIG. 11, we have shown asingle shunt for the input NMs of the interior Λ-MAP with a heavy dashedline. Shunts on the other three NMs are also possible but have beenomitted from the diagram. Connections corresponding to shunts may or maynot be present in the CNS since similar results are possible by learningfrom images transmitted indirectly via the thalamus. Variouscombinations of shunts, and methods equivalent to the formation of thelatch and secondary input associations described above, make it possibleto train the Λ-MAPs in analogous ways.

Referring to FIG. 11, beneath the orthogonal projection operationslabeled V and leading back toward the input and latch NMs is an imagesignal labeled RKE for reproducing kernel error. Unlike the previouserror signals, the RKE signal is not derived from a new data image thatneeds to be remembered, but rather serves to improve other learningprocesses.

Let q and p denote the output of the exterior and interior Λ-MAPmultiplexors respectively. The association relationships established bythe input and latch NMs are at best approximations and will inevitablycontain small errors. Moreover, there is no guarantee that the resultsof a masking operation are valid image spectral representations. For theone-dimensional case, Equation (21) specifies how the reproducing kernelK provides an orthogonal projection V that will map q and p to thenearest functions q and p for which the inverse spectral transformexists. The difference between the input and the output of V is thereproducing kernel error:

RXE _(Q) = q−q= q−V[ q]

RXE _(P) = p−p= p−V[ p]  (49)

Since the projection V is orthogonal, it is idempotent. If the output ofthe multiplexer is already equal to a valid image transform then q=V[q]and the error signal, RKE, will be identically equal to zero.

The RKE signal can help train the input and latch NMs by serving as anerror signal that helps to “fine tune” the NM association formationprocess so that they generate outputs that are more accurate. Inaddition, the formation of new associations may introduce errors intoexisting “old” associations. When recalled, the RKE signal can help to“re-tune” these old associations so the unwanted side effects of the newassociations are eliminated or reduced.

In analyzing the counterparts of Ψ-MAP array model in the nervoussystem, it is essential to distinguish between computational models andphysiological models. Just as an optimizing compiler may modify anequation for better performance, we should expect that the CNS willmodify a computational model to reduce the required number of neuronsand the total expenditure of energy. Mathematically, this correlationcan be expressed as an equivalence class defined by the relation thattwo models are equivalent if for any set of inputs, they always generatethe same outputs. Within this framework, a good computational model isone that is as close as possible to the physiological model, with thecaveat that the computational model may be slightly transformed to makeit easier to understand as long as it remains in the same equivalenceclass.

The vast majority of the synaptic connections in the neocortex arelocal, that is, they arise from neurons that lie within close physicalproximity. The Ψ-MAP design specifically prescribes an interconnectionarchitecture where the PEs have local support.

The neocortex contains billions of neurons continuously processing datain parallel. The intrinsic structure of the Ψ-MAP design specifiesmassive lattices of PEs, all constantly processing data. Moreover, allof the individual Ψ-MAPs within the array can operate concurrently.

The neurons in the cerebral cortex are neatly organized into uniformlayers rather than a large globular mass. There are six cellular layersfound throughout the neocortex. The pattern of interconnections betweenthese six layers is also homogeneous throughout the neocortex. A Ψ-MAPArray model consists of collection of Ψ-MAPs all having the sameorganization of image spectra and the same pattern of interconnection.

Between the cortical layers, many local loops exist where a givenneuron, projects onto other local neurons, which in turn form circuitsor loops that project back onto the original neuron. Referring to FIG.10, the spectral manifolds H, Q, L, P, R, and S are layered one on topof the other. Consequently, the connections between them are local, andform loops analogous to the cortical columns.

Based on the cytoarchitecture, the cerebral cortex is divided into arelatively small number of distinct Brodmann areas. The demarcationbetween the Brodmann areas is determined by the varying prominence orthickness of the six cellular layers. However, within each Brodmannarea, the thickness of the cellular layers is the same. The strikingsimilarity in structure and connectivity implies that all of theBrodmann areas are performing a similar operation.

Each Ψ-MAP in the Ψ-MAP array corresponds to a unique Brodmann area buthas the same general design. The number and source of the inputs and thenumber of associations formed will undoubtedly vary between Ψ-MAPs andeven between the internal and external Λ-MAPs of a single Ψ-MAP. Thesevariations will affect the computational requirements of the differentcortical layers causing them to vary in thickness between Brodmannareas. However, the number of image inputs, and the number ofassociations within any given Brodmann area are the same, suggestingthat within each area the thicknesses of the layers should also be thesame.

Several of the Brodmann areas contain well-documented computational mapsthat correspond to sensory inputs and motor control actions. Thesetwo-dimensional images have a topographic structure that maps to eachparticular sensory modality. During the course of evolution, sensory andmotor control computational maps were presumably some of the earliestimages processed by the CNS. Computational maps have a well-definedtopographic structure, which is an ideal match with the concept ofneural manifolds and image association processors.

The connections between Brodmann areas are topographically organized.That is, during the transfer of information between Brodmann areas,points that are close to each other at the source are also close to eachother at the destination. The transfer of images, or their spectralrepresentations, to other parts of the Ψ-MAP array, requires thepreservation of the underlying topographic structure.

The interface between two neurons occurs at the synapse. Whileelectrical synapses are common in invertebrates, most synapses in themammalian nervous system are chemical, where one or moreneurotransmitters mediate the signal transmission between the pre- andpost-synaptic neurons. While the average concentration of a continuous“cloud” of neurotransmitter representing the value of function definedon a manifold may be the correct model at a high level of abstraction,as a physiological implementation such a cloud would have problems withtemporal inertial and unwanted diffusion. Synapses confine the regionwhere the neurotransmitter is present to the synaptic clefts, therebyrestricting the diffusion and allowing the transmitter to be quicklyreabsorbed or metabolically catalyzed.

In addition to their important function of creating “virtual”neurotransmitter clouds that are more efficient and responsive, thesynapses could also play a critical role in the separation ofvector-valued components. In the Ψ-MAP system equations subsectioncovering multiple inputs, we discussed how Equations (36) and (37)describing the combination of inputs from any number of images could beimplemented with a single type of neurotransmitter by using synapses tomaintain the partition.

In the standard neural network model, a synapse corresponds to a singleweight that represents the effect the presynaptic neuron has on thepostsynaptic neuron. In the neural manifold model, a single layer ofneurons performs the computation of a two-layer neural network where thesecond layer of nodes is the concentration of neurotransmitters in theextracellular space. One set of weights corresponds to the amount ofneurotransmitter released by the cell membrane of the axonal tree andthe second set of weights corresponds to the sensitivity of the cellmembrane of the dendritic tree. In this context, back-propagationalgorithms that set the weights in multilayer neural networks canoperate within the confines of the synaptic regions with errors in theweights corresponding to the dendrite sensitivity propagating back tothe weights corresponding to the axon output.

There is a significant cost to transmit signals along an axon whoselength can be measured on a macroscopic scale. To send the value ofevery single pixel in an image over a relatively long distance wouldrequire an unnecessary expenditure of energy. A more efficient approachis to encode the image spatially so that it can be sent using arelatively small subset of the available neurons. The pyramidal neuronsappear to play a role in the encoding required to transmit images with aminimal consumption of metabolic energy. The nonlinear encodingalgorithm described above in Equation (35) represents an image with onlya few predefined spectral functions. Since the spectral functionscorresponding to nearby pyramidal cells are not orthogonal, a lateralinhibition, equal to the value of the reproducing kernel, allows theselection of a single prominent spectral function in each localneighborhood.

The nonlinear encoding algorithm permits the specification of an errorthreshold that may change over time. Consequently, the nervous systemcan dynamically shift the trade-off between accuracy and the expenditureof metabolic energy to meet the demands of the current situation. Forexample during a fight-or-flight response the CNS may be willing toexpend all of the available energy in exchange for high accuracy, whileduring periods of hunger, or inactivity it may sacrifice accuracy inorder to conserve energy. During these low-energy periods, the imageswould become “dull” or “hazy” and determining the correct associationswould be more difficult and prone to errors.

There may be a physiological correlation between some of the neocorticalcells and the reproducing kernel used the projection operator V. Thereare many types of neurons in the neocortex, including chandelier cells,basket cells, neurogliaform cells, double bouquet cells, and bituftedcells. Any of these cells may be involved in reproducing kernelcomputations, but the chandelier cells in particular have severalrelevant characteristics.

Once the set of spectral functions have been specified, the reproducingkernel, K, is fixed and consequently, the calculations can be done byneurons with a static dendrite or axon response. Furthermore, theprojection V can be computed locally, since if the spectral functionshave local support, so does K. Given its utility in reducing errors (seeEquation (22)), teleonomic arguments imply a projection analogous to Vis likely to take place in the CNS.

Chandelier cells project exclusively to initial portions of axons ofpyramidal cells. They are GABAergic so their effect is to subtract fromthe pyramidal cell's value. The physical extent of the chandelier cellsis limited in the direction perpendicular to the cortical surface sothat they are generally contained within a single layer. This suggeststhat their function is limited to a single spectral manifold and isconsistent with the projection operator V shown in FIG. 10. Also, thereceptive field of the chandelier cells is larger with less acuity thanthe pyramidal cells. This would be expected since for a fixed γ, thereproducing kernel

φ_(α),φ_(γ)

has a larger region of support than the corresponding spectral functionφ_(γ).

In the central nervous system of vertebrates, there are 10 to 50 timesmore glial cells than neurons. Unlike neurons, glial cells do notgenerate action potentials. Consequently, if state is assumed to beencoded in the firing of neurons, glia are relegated to a support role.However, in a neurotransmitter-centric model they can take a centralposition along side neurons. They can participate in both short-term andlong-term memory as well as computations. Without action potentialshowever, the results of the calculations performed by glial cells willbe transmitted more slowly.

To simplify the computational model, each of the spectral manifolds Q,P, H and L receives inputs from only one of the other manifolds.However, there is no a priori reason that requires this limitation. Inparticular, even though is more difficult to mathematically analyze amodel where the NMs receives inputs from multiple sources, it may haveadvantageous performance characteristics that would promote its adoptionduring the course of natural selection.

The neocortex contains two distinct bands of Baillarger, the externalband and internal band. Inputs from the thalamus project mainly to thebands of Baillarger which are affiliated with the overlays T_(Q) andT_(P), used by masking operation M. The bands of Baillarger may also beinvolved in learning from thalamic inputs. Intracortical connectionsalso project to several other strata including the molecular andmultiform layers and modifications to the computational model may berequired to adequately represent these connections.

The nearly simultaneous triggering of even a small fraction of theneurons in a single area will generate a local current densitysufficient to generate a measurable voltage potential on the scalp. Animmediate consequence of the hypothesis that spectral manifolds aresynchronously processed or transferred in the CNS iselectroencephalograms (EEGs). Such variations in neuron triggering rateswill inevitably occur during the transitions of the system clock. WhileEEGs provide evidence of synchronous calculations within the neocortex,this is not to say that EEGs are a system clock, but rather that EEGsresult from the synchronous processing or transfer of entire spectralmanifolds in unison.

Following a certain stage in development, no new neural cells are formedin the cerebral cortex. Learning occurs by extending or otherwisemodifying the existing dendritic trees. After a Ψ-MAP has begun to formassociations, inserting additional untrained PEs into the lattice willintroduce local errors in the manifold calculations and consequentlydegrade the overall performance.

Memories are not lost by the death of a few neurons. Since therelationships are between image spectra, if the associations are storedusing multiple PEs, the removal of some of the PEs from a Ψ-MAP latticewill not result in a noticeable change in its external behavior.

Receptive fields similar to the ones in the visual processing system areknown to exist for the other senses. For example the touching a singlepoint on the surface of the skin is generates a response over an entireregion in the lateral postcentral gyrus. Since associations are betweenthe spectral transforms of images and not the images themselves, sensoryinputs such as retinotopic maps, tonotopic maps and somatotopic mapsmust first be transformed into the distributed spectral representationused in the neocortex before they can be processed.

The CNS is able to integrate the diverse sensations of touch,proprioception, vision, and hearing along with prior memories andknowledge to form a coherent and consistent understanding. Referringback to FIG. 4, it shows how a single PE fuses together regions fromthree separate images. The topographic organization allows any number ofimages to be overlaid and combined as inputs to a single Ψ-MAP that canthen form a coherent and integrated concept.

Visual hyperacuity, the ability to perceive image data at a resolutionthat is higher than the spacing between the rods and cones in theretina, is paradoxical when expressed in terms of neuron actionpotentials. However, by making use of the information that encodes thesmall eye movements or microsaccades, it is possible for aneurotransmitter-cloud model to create and store a visual model at amuch higher resolution.

Short-term memory forms an integral part of our conscious awareness andhas fundamentally different characteristics than long-term memory.Long-term memory is the result of modifying the receptor measures bytraining an NM to associate an output image with one or more inputimages. Short-term memory, on the other hand, is the array of thereciprocal images that are currently holding each other in place in eachof the Ψ-MAPs.

Old memories may become “hazy” or lost. However, “hints” or othercontext can suddenly evoke them, and bring back a very clearrecollection. Within a single Ψ-MAP, all of the PEs are involveddirectly or indirectly in the formation and recollection of eachassociation. If an association is created, but seldom used, it willlikely become “weaker” as other new associations are formed. When thisoccurs, some of the original input patterns may evoke a response that istoo weak to bring back the associated image. However, if a Ψ-MAP hasmany inputs, some small additional context from the other images maysuddenly bring back the entire reciprocal image pair that in turn mayevoke additional details from other parts of the array.

Old and faint memories do not become progressively more blurry as wouldbe expected of an image that is losing resolution. Instead, they retainsome portion of their high-frequency detail. Spectral representationscontain both high and low frequency components. As errors are slowlyintroduced due to of the formation of new memories, the old memorieswill still contain some information across the spectral range.

Even though individual neurons have primarily local connections, enmasse they exhibit emergent properties that lead to a unified, cohesiveglobal behavior. The Ψ-MAP Array architecture specifies how uniformlattice, composed of a massive number of small processing elementsworking in parallel, can achieve an overall, globally integrated result.

We define an “image” or “signal” to be any function defined on amanifold. These may be analog or digital. Examples of signals are audiorecordings, photographs, video recordings, or arbitrary spatialdiagrams.

Each PsyMAP in a PsyMAP Array performs input, processing, and outputoperations. The input process may consist of accepting a signal fromanother PsyMAP, from another intermediate processing device, or from anoutside transducer that performs some type of physical measurement. Theoutput of a PsyMAP is another signal, which is analog in nature, but maycarry digital information the same way an analog voltage signal can beused to transmit digital data. As an example, we can imagine aphotograph as a “distributed bar-code” where we identify multipleregions of the image as part of a single “bar” and use the average ofpixel values taken within the regions to make binary assignments. Wecould define average values between 0.0 to 0.2 equal to “zero” andaverage values between 0.8 to 1.0 equal to “one”. In this manner,multiple regions in a PsyMAP output signal or image can be equated withan arbitrary ASCII character or character string. Consequently, a“Manifold Association” process is also a signal recognition process thatcan be used to identify, classify, or analyze any input signal and havethe output interface with any discrete digital device.

Fundamentally, the PsyMAP methods are mathematical expressions that canbe evaluated by a variety of digital and analog devices. While the useof optical computers may be possible, we expect that typicalimplementations will be done on a digital computer. Because of thenumber of possible trade-offs between image resolution, processingpower, processing time, association capacity, association accuracy, andthe number of PsyMAPs in the PsyMAP Array, it is possible to execute thealgorithms on hardware ranging from handheld devices to the largest ofsupercomputers. Applications suitable for small consumer devices mightinclude handwriting recognition, while more powerful machines will berequired for large image data sets and fast throughput. The results ofthe analysis may be stored onto a volatile or non-volatile storagemedium such as a hard disk drive, a flash memory, or other suitablestorage medium, and the results of the analysis may be displayed on thecomputer CRT (terminal), any other display device, printed out on paper,or printed on any other suitable medium.

The PsyMAP algorithms are inherently parallelizable and consequently canbe implemented either by a single processor or by a large number ofprocessors. Parallel implementations can be constructed using multipleprocessors that operate on different (but somewhat overlapping) sectionsof the same image, as well as assigning the processors to differentimages within the PsyMAP Array.

The following examples illustrate different applications of the presentinvention:

Example 1 Speech Recognition

An exemplary application is a speech recognition device for a personalcomputer (PC). A microphone connected to the PC provides an input signalto a manifold array processor, while the output is the stream ofcharacters corresponding to the words spoken by the user. The incomingaudio signal first undergoes frequency analysis pre-processing such as awindowed Fast Fourier Transform (FFT). This would be followed by a Ψ-MAPclassifier/recognizer and then followed by post-processing to write theΨ-MAP output to a character stream. These calculations could beperformed by the main CPU (central processing unit) of the PC itself oron a small separate “daughter” card attached as a peripheral to the maincircuit board. The daughter card may contain a single microprocessor ormultiple processors and a combination of one or more programmabledigital signal processing (DSP) chips.

Example 2 Pen Stroke Analysis

A handheld Personal Digital Assistant (PDA) device may have atouch-sensitive screen that records the pressure of pen strokes on itssurface. The recorded result (x(t),y(t)) is a vector valued functiondefined on a one-dimensional manifold (time). A pre-processing step mayscale or normalize the signal by performing arc-length encoding ortranslating it so that it is centered about its two-dimensionalcenter-of-gravity. The signal may then be used as input to severalΨ-MAPs within a Ψ-MAP array, which analyze it for the presence ofvarious characteristics such as corners or intersections. The outputs ofthe individual Ψ-MAPs may then be topologically aligned as inputs to afinal Ψ-MAP stage using the one-dimensional equivalent of thetwo-dimensional alignment shown in FIG. 4. This last Ψ-MAP stage in thearray may then recognize the topologically aligned inputs and produce asingle signal corresponding to a waveform that encodes the bits of anASCII character that was drawn on the screen of the PDA. Thesecharacters can then be used anywhere that a keyboard input wouldnormally be required, such as a word processing program or textmessaging application.

Example 3 Two-Dimensional Image Encoding

In some applications, the input is an image, for example, from a CCDcamera or imaging array, which generates a digital signal that is thenassociated with an encoded image, such as a bar code or characterstream, where the code may represent a word that identifies or describesthe object within the image. Generally, the present invention isapplicable to any image or signal that can encode digital data, such asface recognition, handwriting recognition, object recognition, etc. In aface recognition application, a person's social security number can beassociated with a bar code in which is encoded the person's image. Itshould be noted that bar code is used as an example, and other symbols,including one or more alphanumeric characters may be used to encode theimage or signal.

Example 4 Image Encoding—Histology

The collection of images from a single tissue sample taken under amicroscope at high magnification represents a huge data set that isdifficult and tedious for a human operator to analyze. Searching thisdata set for diseased or cancerous cells is a critical step in medicaldiagnosis.

Various types and shapes of convolution kernels can be used to estimategradients, edges and textures in an image. Symmetric and asymmetrickernels may form the frame functions for one Ψ-MAP in a Ψ-MAP Array andthe corresponding output images may be used as inputs or masks toanother Ψ-MAP. The topological alignment of gradient, texture and colorinputs may be used to identify individual cells and cell boundaries. Theoutput of this “cell-identification” stage may be a mask that equalszero outside the cell and equals one inside. This mask is then used by aclassification Ψ-MAP that uses texture and color information to classifythe cell type. A stable reciprocal image pair within a single Ψ-MAPdefines each class with one of the images corresponding to a “bar-code”or other binary encoding of the class identifier. Multiple classes wouldexist for various types of healthy and diseased cells. The output ofsystem would be a written report that is read by a physician or medicaltechnician. The report may contain overall statistics on cell types aswell as sample images of suspected pathogens or cancerous cells andtheir estimated boundaries.

Since this histology application would be incorporated into a systemcontaining costly microscopy equipment and would probably be used in ahigh-throughput specimen processing facility, it will likely be costeffective to construct it using a large multiprocessing system withhundreds or even thousand of CPUs.

Example 5 Handwriting Recognition

Handwriting recognition has an output similar to the “pen-strokeanalysis” described above, but the input is a scanned two-dimensionalimage rather that the one-dimensional path describing the motion of apen in time. Known image processing techniques can generaterepresentations that are invariant under translation, scale and rotationoperations. These transformations in combination with Ψ-MAP arrays canbe used to read and process handwritten forms, letters or diagrams.

Example 6 Three-Dimensional Shape Analysis—Oncology

Magnetic Resonance Imaging (MRI) and Computed Tomography (CT) arecommonly used medical diagnostic devices that provide three-dimensionalvolumetric data. The same Ψ-MAPs array theory described fortwo-dimensional images works equally well in three or more dimensions.For MRI and CT processing, we may use three-dimensional sphericallysymmetric wavelets. The resulting spectral manifold is thenfour-dimensional. For an oncology application, the patient image data issearched for the presence of tumors. One on the Ψ-MAPs in a Ψ-MAP arrayhas two stable states corresponding to “Tumor” and “Non-Tumor”. Thesetwo states could be represented by all white and all black images in oneof the two Ψ-MAP outputs. Whenever this Ψ-MAP switches to the “tumor”state the position and characteristics are recorded and included in areport sent to a medical technician or radiologist. The Ψ-MAP is thenreset and the scanning of the dataset continues until completed.

Example 7 Three-Dimensional Face Recognition

Several techniques such as stereoscopic reconstruction and laserscanning are available which automatically construct a representation ofthe surface of a three-dimensional object. The representations may bevolumetric (voxel) or polyhedra that can be easily converted to voxels.In a voxel representation, a voxel has a value of one if the cube iscompletely inside the object and a value of zero if it is outside.Voxels on the surface have a value proportional to the percentage of thecube inside the object. The recognition can be accomplished using thethree-dimensional version of the two-dimensional image encodingdescribed above. Since the bone structure of the face is fixed, once thedata has been aligned with the sagittal plane of symmetry, facerecognition based on three-dimensional structure is more reliable thantwo-dimensional photographic techniques that can be easily thrown off bymakeup, eyeglasses, and facial hair. Using the measures defined forneural manifold pattern associations, the recognition process may beadjusted so that some regions of the face have greater importance in thesimilarity calculations.

Example 8 Hand Gesture/Sign Language Recognition

A video signal is a two-dimensional image changing in time and cantherefore be regarded as a function defined on a three-dimensionalmanifold. Similarly, a volumetric (voxel) representation of a personspeaking in sign language is a function defined on a four-dimensionalmanifold (3D space×time). Using the image encoding “bar-code” techniquedescribe above, we can associate a finite set of signs or gestures witha finite set of digital codes. In this manner, a Ψ-MAP array gesturerecognition device could be interfaced with a text application in thesame way as a speech recognition device.

Example 9 Industrial Inspection

An imaging device may be positioned over an assembly line or conveyorbelt that causes parts or materials to come into view. Multiple Ψ-MAPwithin an array may analyze the object in terms of variouscharacteristics such as shape, texture, and color. Imperfections may bedetermined not just as discrete individual tokens, but on a continuousscale measured in degrees of severity and described by a functiondefined on a manifold that is “overlaid” on the actual object. One ofthe Ψ-MAPs, which directly or indirectly accepts inputs form all of theother Ψ-MAPs in the array, would be bi-stable with two specialpredefined output images corresponding to “Accept” or “Reject”. Thisbinary output would then be sent to the assembly line control mechanism.

Example 10 Noise Removal

If the content of a signal is known a priori to come from a restrictedset of inputs, then recognizing which particular element is present andusing that information to create a reconstructed output can have theeffect of reducing noise. For example, if all of the fonts and fontsizes used to produce a document are known, recognition of thecharacters and there positions can be used to produce a new documentthat is relatively free of background noise. The effect is the same asif a human operator had re-entered text from an old and severelydegraded printed document.

Application Encodings

The commonality in all of these examples is that encoding ortransforming functions has many useful purposes. These often result froma many-to-one mapping that takes a infinite variety of different inputfunctions and maps them to a smaller, perhaps finite, set of outputfunctions. The Ψ-MAP Array methods combine several characteristics thatallow it to be used in a very wide range of applications. Theseinclude: 1) a new neuron model that makes learning associations easierand faster, 2) the ability to specify frame functions that define alinearly transformed space that is best suited to the particularsub-task, 3) the ability to combine and integrate any number of inputsfrom external sources or other PsyMAPs and 4) an orthogonal projectionbased on the reproducing kernel that uses redundancy information toreduce noise and increase stability in the reciprocal images. Together,these capabilities create a very flexible and powerful functiontransformation system with many uses and applications.

The foregoing description of preferred embodiments is not intended to belimited to specific details disclosed herein. Rather, the presentinvention extends to all functionally equivalent structures, methods anduses as may fall within the scope of the appended claims.

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1. A method for generating a spatially-encoded output signal from aninput signal having a structure comprising a plurality of discretesignal units, the method comprising: inputting the input signal into acomputer processor having software stored therein for executing aplurality of processing elements, wherein each processing elementcomprises a logic manifold association processor having an input and anoutput, and wherein each processing element generates a single outputsignal; transforming the input signal into representations defined onmanifolds of one of a physical space, a projective geometry, andtopological manifolds of rotations or translations; forming a recursiveloop between a pair of processing elements, wherein the output of eachprocessing element of the pair of processing elements is fed back intothe input of the other processing element of the pair, and wherein atleast one of the pair of processing elements further receives an elementinput comprising a small subset of the plurality of discrete signalunits of the input signal; training a plurality of pairs of processingelements to detect and form associations between a plurality ofreciprocal signal pairs, wherein when a first signal of a reciprocalsignal pair of the plurality is input into a first processing element,the first processing element outputs a second signal of the reciprocalsignal pair, and when the second signal of the reciprocal signal pair isinput into a second processing element, the second processing elementoutputs the first signal, so that the reciprocal signal pair convergesto a fixed signal that is locked within the pair of processing elements;wherein when the at least one of the pair of processing elements thatreceives the small subset of discrete signal units receives the inputsignal units, the two processing elements generate an encoded outputcomprising spatial information associated with the subset of inputsignal units; and storing the encoded output in a memory incommunication with the computer processor.
 2. The method of claim 1,wherein the discrete signal units within the input signal havecorresponding spatial locations that are encoded with a frequencyrepresentation, wherein each processing element encodes a different gridof spatial locations.
 3. The method of claim 1, wherein the input signalfurther comprises temporal knowledge where a product manifold is formedwith the spatial manifolds and a time manifold, or a frequencyrepresentation of time, the resulting knowledge encoding eventscorresponding to a unique time and place.
 4. The method of claim 1,wherein the input signal is a three-dimensional image and the step oftransforming comprises aligning the image with a plane of symmetry.